 •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
3.4 Exercises 
55 
undecidable. For neither theoremhood nor tautologousness in classical predicate logic is it possible to devise a mechanical test that is guaranteed to yield, for any formula, a correct yesorno classiﬁcation in a ﬁnite number of steps. On the other hand, ﬁrstorder tautologousness (and hence theoremhood) is semidecidable: there are for example mechanical tests based on resolution or semantic tableaux that will, after a ﬁnite number of steps, always yield a correct yes classiﬁcation for any ﬁrstorder formula that is a tautology. But such tests do not yield full decidability because they may fail to yield any answer within a ﬁnite number of steps for formulas that are not tautologies.7
3.4 Exercises
SECTION 3.2
1Determine the truthvalue of each of the following formulas on an interpretation that makes the following assignments:
D:set of positive integers
I(O) = {<u>: u D and u is odd}
I(S) = {<u1, u2>: u1 D, u2 D, and u1 squared is u2}
I(E) = {<u1, u2>: u1 D, u2 D, and u1 evenly divides u2}
I(a) = 1
I(b) = 2
I(c) = 3
a.Oa Oc
b.Oa → Ob
c.( x)Ox
d.( x)Exx
e.( x)¬Sxx
f.( x)Sxb
g.( x) ( y)((Ox Oy) → Exy)
h.( x)( y)(Exy Eyx)
i.( x)( y)(¬Ox Exy)
j.( x)( y)(Sxy → Exy)
k.( x)( y)(Sxy Syx)
l.( x)(Ox → ( y)( z)(Sxy Syz))
m.( x)( y)Sxy
n.( y)( x)Sxy
7Church (1936) proved that theoremhood in ﬁrstorder logic is undecidable. See Part 4 of Hunter (1971) for a general proof of undecidability in various ﬁrstorder systems. A semi–decision procedure for ﬁrstorder classical logic based on semantic tableaux is presented in Smullyan (1968, pp. 59–60).
56 
Review of Classical FirstOrder Logic 
SECTION 3.3
2a. Prove that every instance of the axiom schemata CL 4 is a tautology in classical ﬁrstorder logic.
b.Prove that every instance of the axiom schemata CL 5 is a tautology in classical ﬁrstorder logic.
3Construct a derivation that justiﬁes derived axiom schema CL D7.
4Construct a derivation that shows that ¬( x) (¬Tx ¬¬Tx) is a theorem of CL A. You will ﬁrst need to use the deﬁnitions of the existential quantiﬁer and conjunction to write the formula without those operators.
5Show that the following rule can be derived in CL A:
EG (Existential Generalization). From P(a/x) infer ( x)P where x is any individual variable
6Construct derivations to show that the following formulas are theorems of CL A (rewriting each formula to obtain a formula that contains only negation, the conditional, and the universal quantiﬁer as operators):
a.( x)( y)Lxy → ( x)( y)Lxy
b.( x)(Fa → Gx) → (Fa → ( x)Gx)
c.( x)( y)Lxy → ( y)( x)Lxy
d.( x)( y)Lxy → ( x)Lxx
4 Alternative Semantics for TruthValues and TruthFunctions: Numeric TruthValues and Abstract Algebras
4.1 Numeric TruthValues for Classical Logic
We’ve been using the letters T and F to stand in for the truthvalues true and false. We could just as well use the numerals 1 and 0 to stand for true and false, recasting truthtables using these two numerals, for example,
P 
¬P 
1 
0 
0 
1 
More interesting, though, is using the integers 1 and 0, rather than the numerals that name these integers, as the truthvalues of formulas in classical logic. A propositional truthvalue assignment will consist in the assignment of one of these values to each atomic formula, and we can then numerically deﬁne the values for complex formulas rather than simply listing these values in truthtables. Letting V(P) mean the value of P on a (numeric) truthvalue assignment V, the following deﬁnitions will do the job:
1.V(¬P) = 1 – V(P)
2.V(P Q) = min (V(P), V(Q)) (i.e., the minimum of these two values)
3.V(P Q) = max (V(P), V(Q)) (i.e., the maximum of these two values)
4.V(P → Q) = max (1–V(P), V(Q))
5.V(P ↔ Q) = min (max (1–V(P), V(Q)), max (1–V(Q), V(P)))
Clause 1 “reverses” the value for a negated formula—since (looking at the righthand side of the formula) 1 − 1 is 0 and 1 − 0 is 1. Clause 2 indicates that a conjunction is only as true as its least true conjunct. By contrast, clause 3 indicates that a disjunction is as true as its most true disjunct. Clause 4 is based on the equivalence of P → Q and ¬P Q, using clauses 1 and 3 to deﬁne the truthconditions for the latter. Clause 5 is based on the equivalence of P ↔ Q and (P → Q) (Q → P). If we had instead used the equivalence of P ↔ Q and (P Q) (¬P ¬Q) to capture the truthconditionals for biconditionals, the righthand side of clause 5 would have been written as max (min (V(P), V(Q)), min (1–V(P), 1–V(Q))). This is equivalent to the righthand side that we have chosen to use for clause 5 (and is left as an exercise to verify).
57
58 
Alternative Semantics for TruthValues and TruthFunctions 
Clauses 1–5 produce truthtables that look exactly like our previous classical truthtables, except that we now have 1 in place of T and 0 in place of F:
P 


¬P 



























1 


0 














0 


1 














P Q 

P Q 

P Q 

P Q 

P Q 

P → Q 

P Q 

P ↔ Q 









1 1 


1 
1 1 

1 
1 1 

1 
1 1 

1 

1 0 


0 
1 0 

1 
1 0 

0 
1 0 

0 

0 1 


0 
0 1 

1 
0 1 

1 
0 1 

0 

0 0 


0 
0 0 

0 
0 0 

1 
0 0 

1 
Note that conjunction could equivalently be deﬁned in terms of multiplication as V(P Q) = V(P) · V(Q). This will become important when we turn to fuzzy logic, where truthvalues are always deﬁned numerically. And disjunction is almost like addition, except in the case where both disjuncts have the value 1, so an alternative clause for disjunction is V(P Q) = min (1, V(P) + V(Q)).
Having substituted 1 and 0 for true and false in clauses deﬁning the values of complex formulas, we need to make the same substitutions in other semantic deﬁnitions: a tautology of classical propositional logic is a formula that always has the value 1; a contradiction is a formula that always has the value 0; an argument is valid if its conclusion has the value 1 whenever its premises have the value 1; and so on. Thus we will arrive at the same set of tautologies as we had based on the values true and false, and other semantic results also remain the same. The difference is that now we can compute values for formulas numerically rather than simply referring to truthtables. For example, we can show that A ¬A is a tautology of classical propositional logic as follows: V(A ¬A) = max (V(A), 1–V(A)), and when V(A) = 1 the maximum is V(A), that is, 1, while when V(A) = 0 the maximum is 1 – V(A), that is, 1. Thus the formula always has the value 1 and is therefore a tautology.
We can similarly deﬁne numerical values for formulas of classical ﬁrstorder logic. Instead of talking about a variable assignment satisfying or not satisfying a formula, we may talk instead about formulas having the value 1 or 0 on variable assignments. We’ll designate the value that a formula P has on a variable assignment v on an interpretation I with the notation Iv(P):
1.Iv(Pt1 . . . tn) = 1 if <I*(t1), . . . ,I*(tn)> I(P), where I*(ti) is I(ti) if ti is a constant and is V(ti) if ti is a variable, and IV(Pt1 . . . tn) = 0 otherwise.
2.Iv(¬P) = 1 – Iv(P).
3.Iv(P Q) = min(Iv(P), Iv(Q))
4.Iv(P Q) = max(Iv(P), Iv(Q))
5.Iv(P → Q) = max(1 – I(P), Iv(Q))
6.Iv(P ↔ Q) = min(max(1 – Iv(P), Iv(Q)), max(1 – Iv(Q), Iv(P)))
7.Iv(( x)P) = min{Iv (P): v is an xvariant of v}
8.Iv(( x)P) = max{Iv (P): v is an xvariant of v}