 •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
32 
Review of Classical Propositional Logic 
and we rewrite the ﬁrst premise as E → (¬U → S). Here is a derivation:
1 

E → (¬U → S) 
Assumption 



2 

U → W 
Assumption 

3 

S → W 
Assumption 

4 

¬W 

Assumption 
5 

¬U 
2,4 MT 

6 

¬S 
3,4 MT 

7 

(¬U → S) → (¬S → ¬¬U) 
CLD5, with ¬U / P, S / Q 

8 

¬S → ((¬U → S) → ¬¬U) 
7, TRAN 

9 

(¬U → S) → ¬¬U 
6,8 MP 

10 

((¬U → S) → ¬¬U) → (¬¬¬U →¬ (¬U → S)) 
CLD5, with ¬U → S / P, ¬¬U / Q 

11 

¬¬¬U → ¬ (¬U → S) 
9,10 MP 

12 

¬U → ¬¬¬U 
CLD4, with ¬U / P 

13 

¬¬¬U 
5,12 MP 

14 

¬(¬U → S) 
11,13 MP 

15 

¬E 
1,14 MT 
We stress that these derived axiom schemata and rules are a convenience for constructing derivations; the set of axiom schemata CL1–CL3 alone with the single rule MP form a complete derivation system for classical propositional logic, and so the additional axioms and rules do not add to the power of the system. Nor do they affect its soundness, since they are all derivable within a system that was sound to begin with.
2.5 Functional Completeness
In Section 2.2 we pointed out that we could take one of several pairs of connectives as primitive and deﬁne the rest in terms of these. In Section 2.4 we took advantage of this fact: the axiomatic system CLA uses only two connectives, since formulas containing the other connectives can be rewritten using the two connectives ¬ and → that appear in CLA.
There is an important related theoretical issue to which we now turn. First, we formally deﬁne the concept of a truthfunction: a truthfunction is a function that maps truthvalues to truthvalues. More speciﬁcally, a truthfunction is always a function of a given ﬁnite number of arguments: one, two, three, whatever; it is a function that maps each sequence of the appropriate number of truthvalues to a truthvalue. The negation truthfunction is a truthfunction of one argument as speciﬁed by the following truthtable template:
T F
F T
2.5 Functional Completeness 
33 
This function maps the single truthvalue T (more precisely, the singlemembered sequence <T>) tothe truthvalue F,and it maps the single truthvalue F (<F>) to the truthvalue T. The conditional truthfunction is a truthfunction of two arguments:
T T 
T 
T F 
F 
F T 
T 
F F 
T 
It maps the sequence <T, F> to the truthvalue F, and all other sequences of two truthvalues to the truthvalue T.
We say that a formula P of propositional logic expresses a truthfunction of n arguments if the truthtable for P speciﬁes that truthfunction; that is, the values under P’s main connective are the values to which the function maps each sequence of n truthvalues listed to the left of the vertical line. So, for example, and by design, ¬P expresses the negation truthfunction, and P → Q expresses the conditional truthfunction (other formula letters may be used). Other truthfunctions may require more complicated formulas. For example, the neithernor truthfunction, captured in the following truthtable template,
T T F
T F F
F T F
F F T
is expressed by the formula ¬(P Q) or equivalently by ¬P ¬Q.
Here is the theoretical issue: can every classical truthfunction be expressed by a formula of classical propositional logic using only the ﬁve connectives ¬, ,, →, and/or ↔? The answer is yes. We shall show how, given any truthfunction, to construct a formula that expresses exactly that truthfunction. To facilitate the proof we shall assume that the function in question has been laid out in a truthtable template as previously, and that n is the number of arguments that the truthfunction operates on.
First, we choose n atomic formulas P1, . . . , Pn, one corresponding to each argument place. These will head the columns to the left of the vertical line in the truthtable template. Next, for each row of the truthfunction template we form a corresponding conjunction conjoining the atomic formulas that have the value T in that row along with the negations of the atomic formulas that have the value F—in the terminology of Section 2.3, each such conjunction is a phrase. So, for example, phrases corresponding to the four rows of the neithernor function template are, respectively, P Q, P ¬Q, ¬P Q, and ¬P ¬Q. Note that each of these phrases is true exactly when P and Q have the truthvalues in its corresponding row. Next we form a disjunction of the phrases corresponding to the rows that have T to the right of the vertical line, thus producing a formula in disjunctive normal form. In the case of the neithernor function there is one such row, the fourth, so we form the
34 
Review of Classical Propositional Logic 
“disjunction” of the single phrase for that row: ¬P ¬Q. (A “disjunction” of a single formula is simply the formula.) This formula expresses the function captured in the neithernor truthtable template.
As a more complicated example consider the function of three arguments:
T T T 
T 
T T F 
T 
T F T 
T 
T F F 
F 
F T T 
F 
F T F 
F 
F F T 
T 
F F F 
F 
Assuming that we have chosen the atomic formulas P, Q, and R, a disjunction of phrases corresponding to the rows with T to the right of the vertical line is
((((P Q) R) ((P Q) ¬R)) ((P ¬Q) R)) ((¬P ¬Q) R). This formula expresses the truthfunction speciﬁed in the truthtable template, as the reader may easily conﬁrm.
Wemust add two special cases. In one we have a truthfunction of one argument, such as,
T T
F T
In this case the phrase corresponding to a row is a single atomic formula or its negation: P for the ﬁrst row and ¬P for the second. Since both rows contain T to the right of the vertical line, the disjunction P ¬P of these two phrases expresses the truthfunction. In the other case there are no Ts to the right of the vertical line. In this case we may simply conjoin the phrase corresponding to the ﬁrst (or any other) row with its negation. So for the truthfunction
T F
F F
we have the formula P ¬P, and for the truthfunction
T T F
T F F
F T F
F F F
we have the formula (P Q) ¬(P Q).
Note that there are other formulas—in fact inﬁnitely many other formulas— that express these same truthfunctions, so it is important to keep in mind that we are only showing that, given any truthfunction, there is at least one formula using the ﬁve connectives that expresses it. Now, we’ve claimed that our procedure will
2.6 Decidability 
35 
always work—but how do we know this? It’s rather simple. Each phrase corresponding to a row of a truthfunction template is true on the truthvalue assignments represented by that row that is false on all other truthvalue assignments. Thus a disjunction of the phrases corresponding to rows that have a T to the right of the vertical line will be true on the truthvalue assignments represented by those rows and false on all other truthvalue assignments. In the case where there are no Ts to the right of the vertical line we produce a contradictory formula of the general form P ¬P, which is always false. Conclusion: we’ve speciﬁed a way to construct, for any classical truthfunction, a formula that exactly expresses that truthfunction.
When a set of connectives is sufﬁcient to express every truthfunction, we say that the set of connectives is functionally complete. Thus, we have proved
Result 2.6: The set of connectives {¬, , } is functionally complete,
since these are the only connectives we have used in formulas to express any truthfunction. From this it follows that the full set {¬, , , →, ↔} is also functionally complete—since we are not required to use all of the connectives in the candidate formulas expressing the various truthfunctions. But since we know that there are three subsets consisting of only two of our connectives that are sufﬁcient to deﬁne the others, we may also conclude that those three subsets, {¬, }, {¬, }, and {¬, →}, are truthfunctionally complete.9
2.6 Decidability
Classical propositional logic has a desirable property that isn’t shared by all logical systems: its set of tautologies is decidable. A set of formulas is decidable if there is a decision procedure for membership in the set, that is, a mechanical procedure that will, given any formula, correctly decide after a ﬁnite number of steps whether that formula is a member of .10
The set of tautologies of classical logic is decidable because there exist mechanical procedures for testing whether a formula is a tautology. We’ve already seen one such procedure: given any formula we can construct a truthtable for that formula and examine the column of truthvalues under the formula’s main connective. If that column consists solely of Ts then the formula is a tautology; otherwise it is not. Clearly truthtables can be constructed mechanically, and just as clearly the construction and examination of the relevant column of truthvalues take only a ﬁnite number of steps. Similarly, the set of contradictions of classical propositional logic
9These are the only sets consisting of two of the ﬁve connectives that are truthfunctionally complete. First, we note that we need the negation connective, for without it we can never produce a formula that is false when all of its atomic components are true. Second, we note that negation and the biconditional won’t sufﬁce because, for example, every formula constructed from two atomic formulas using only these two connectives will have an even number of Ts and an even number of Fs in its truthtable (proof is left as an exercise).
10Sets that contain things other than formulas can also be said to be decidable, but that is not our interest here.