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13.6 Summary: Comparison of FuzzyL, FuzzyG, and FuzzyP
and Their Derivation Systems
FuzzyL
FuzzyG
FuzzyP
In this section we place various results about the three major fuzzy propositional logics in a table, for overall comparison:
|
|
Weakly |
Tautologies/ |
Strongly |
Fuzzily |
Semantically |
Deduction |
complete |
theorems |
complete |
complete |
compact |
theorem |
axiomatization |
decidable |
axiomatization |
axiomatization |
No |
Modified |
Yes |
Yes |
No |
Yes |
Yes |
Yes |
Yes |
Yes |
Yes |
No |
No |
Modified |
Yes |
Yes |
No |
No |
(The set of theorems for each of FLPA, BLLA, BLGA, and BLPA is decidable, as a consequence of the decidability of the tautologies of FuzzyL, FuzzyG, and FuzzyP as noted in Chapter 11).
Recalling that the strong completeness of an axiomatic system is tied to semantic compactness, we can characterize the major differences as follows: FuzzyG is the only one of the three systems that is semantically compact and in which the standard Deduction Theorem holds, while FuzzyL is the only one of the three for which an adequate Pavelka-style axiomatization is possible. As a consequence of this latter fact, and the strong fuzzy completeness of FLPA, we also have fuzzy compactness for FuzzyL.
13.7 External Assertion Axioms
Because external assertion is not definable in any of our three fuzzy logics, none of the the axiom systems we’ve examined is sufficient for derivations involving this operation. Matthias Baaz (1996) formulated the following axioms, which may be added either to FLA (and FLPA)15 or to BLA (and any system on which BLA is based), for external assertion:
∆1. ∆P ¬∆P
∆2. ∆(P Q) → (∆P ∆Q)
∆3. ∆P → P ∆4. ∆P → ∆∆P
∆5. ∆(P → Q) → (∆P → ∆Q)
15But with an important caveat in this case: although FLPA with these axioms and rule will be fuzzy sound for FuzzyFL augmented with the external assertion operation, it will no longer be (strongly) fuzzy complete. Again, the problem is that external assertion is not a continuous operation.
It may strike the reader as unusual that we will actually produce derivations in the incomplete FL PA—knowing that it is not a fully adequate system. We do so not only to illustrate the external assertion axioms and rules but also to demonstrate the potential usefulness of an axiomatic system that isn’t complete. The usefulness stems from the fact that there is no decision procedure for n-tautologousness or n-degree-entailment in FuzzyL (the former is proved in Hajek´ (1995a) but sometimes a derivation might be quite simple to produce.
13.7 External Assertion Axioms |
255 |
We also add the rule
EA (External Assertion). From P infer ∆P.
We will denote derivation systems augmented with these axiom schemata and the rule EA by adding the symbol to the names of those systems, for instance, FL A.
In Section 11.9 of Chapter 11 we noted that axiom schema 1 is a tautology in FuzzyL augmented with the external assertion operator. By including axiom schema1 it becomes a theorem of FL A. The converse of axiom schema 3 is not included as an axiom because it is not a tautology in any of our fuzzy systems. If P has the value .5, for example, then P → ∆P has the value .5 in FuzzyL (augmented with ) and the value 0 in FuzzyG and FuzzyP.
We also noted in Section 11.9 that if a formula P has any value other than 1 or 0, then the formula ¬∆P ¬∆¬P is true in FuzzyL. Here’s a derivation in FL PA that shows that when the formula P has the value .5, ¬ P ¬ ¬P has the value 1:
1 |
[1/2 → P, 1] |
Assumption |
2 |
[P → 1/2, 1] |
Assumption |
3 |
[ P → P, 1] |
3, with P / P |
4 |
[ P → 1/2, 1] |
2,3 HS |
5 |
[¬1/2 → ¬ P, 1] |
4, GCON |
6 |
[1/2 → ¬1/2, 1] |
FLP6.2 |
7 |
[1/2 → ¬ P, 1] |
5,6 HS |
8 |
[ P → ¬ P, 1] |
4,7 HS |
9 |
[( P → ¬ P) → ¬ P, 1] |
1, with P / P |
10 |
[¬ P, 1] |
8,9 MP |
11 |
[¬P → ¬1/2, 1] |
1, GCON |
12 |
[¬1/2 → 1/2, 1] |
FLP6.1 |
13 |
[¬P → 1/2, 1] |
11,12 HS |
14 |
[ ¬P → ¬P, 1] |
3, with ¬P / P |
15 |
[ ¬P → 1/2, 1] |
13,14 HS |
16 |
[¬1/2 → ¬ ¬P, 1] |
15, GCON |
17 |
[1/2 → ¬ ¬P, 1] |
6,16 HS |
18 |
[ ¬P → ¬ ¬P, 1] |
15,17 HS |
19 |
[( ¬P → ¬ ¬P) → ¬ ¬P, 1] |
1, with ¬P / P |
20 |
[¬ ¬P, 1] |
18,19 MP |
21 |
[¬ P ¬ ¬P, 1] |
10,20 WCI |
As one more example in FL PA, we’ll show that if the value of P is at most .5, then ¬ P is true. The reasoning is that if P has at most the value .5, then P has at most the value .5 (line 3), and it follows from the disjunctive axiom schema 1 that ¬P must be true. The latter is established by first deriving the formula on line 9, then showing that it follows from this formula that ¬P holds. In the derivation we switch
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Derivation Systems for Fuzzy Propositional Logic |
freely between the equivalent forms P Q and (P → Q) → Q, depending on whether we are viewing the formula as a disjunction or as a conditional:
1 |
|
P → 1/2 |
|
Assumption |
|
|
|||
2 |
|
P → P |
3, with P / P |
|
3 |
|
P → 1/2 |
1,2 HS |
|
4 |
|
1/2 → (¬ P 1/2) |
FLP1, with 1/2 / P, ¬ P → 1/2 / Q |
|
|
|
{Note: ¬ P 1/2 is defined to be (¬ P →1/2) →1/2} |
||
5 |
|
P → (¬ P 1/2) |
3,4 HS |
|
6 |
|
¬ P → (¬ P 1/2) |
FLPD3, with ¬ P / P, 1/2 / Q |
|
7 |
|
( P ¬ P) → (¬ P 1/2) |
5,6 DC |
|
8 |
|
P ¬ P |
1, with P / P |
|
9 |
|
(¬ P → 1/2) → 1/2 |
7,8 MP |
|
10 |
|
((¬ P → 1/2) → 1/2) → ((1/2 → ¬ P) → ¬ P)) |
FLP4, with ¬ P / P, 1/2 / Q |
|
11 |
|
((1/2 → ¬ P) → ¬ P)) |
9,10 MP |
|
12 |
|
¬1/2 → ¬ P |
3, GCON |
|
13 |
|
1/2 → ¬1/2 |
FLP6.2 |
|
14 |
|
1/2 → ¬ P |
12,13 HS |
|
15 |
|
¬ P |
11,14 MP |
|
13.8 Exercises
SECTION 13.1
1Show that the following are derivable as rules in the axiomatic system FLA for FuzzyL (you will probably find it useful to refer to proofs in Chapter 6 as a guide, but recall that L34 is not an axiom schema in FLA):
a.CON. From ¬P → ¬Q infer Q → P.
b.MT. From ¬P and Q → P derive ¬Q.
c.LSIMP. From P Q infer P.
d.RSIMP. From P Q infer Q.
e.SUB. From P → Q, Q → P and a formula R that contains P as a subformula, infer any formula R* that is the result of replacing one or more occurrences of P in R with Q.
f.DN. From any formula R that contains P as a subformula, infer any formula R* that is the result of replacing one or more occurrences of P in R with ¬¬P, and vice versa.
g.TRAN. From any formula R that contains P → (Q → S) as a subformula, infer any formula R* that is the result of replacing one or more occurrences of
P → (Q → S) in R with Q → (P → S).
h.GCON. From any formula R that contains P → Q as a subformula, infer any formula R* that is the result of replacing one or more occurrences of P → Q in R with ¬Q → ¬P, and vice versa.
13.8 Exercises |
257 |
i.GHS. From a conditional (P1 → (P2 → (P3 → . . . (Pn−1 → Pn) . . . ) and Pn → Q, infer (P1 → (P2 → (P3→ . . . (Pn−1→ Q) . . . ).
j.GMP. From a conditional (P1 → (P2 → (P3 → . . . (Pn−1 → Pn) . . . ) and one of the antecedents Pi, 1 ≤ i ≤ n−1, infer the conditional that results from deleting Pi, the conditional arrow following Pi, and associated parentheses
k.DS. From P Q, P → R and Q → R, infer R.
2Explain why we do not want the following as a derived rule in FLA (although it is derivable in L3A):
From P → Q and (P → ¬P) → Q, infer Q.
3Show that the following are derivable as axiom schemata in FLA (similar hint to that in Exercise 1):
a.FLD10. ((P → P) → Q) → Q
b.FLD11. ¬(P → Q) → P
c.FLD12. ¬(P → Q) → ¬Q
d.FLD13. (P → Q) (Q → P)
4Show that the following are derivable as rules in FLA:
a.L&SIMP. From P & Q, infer P.
b.R&SIMP. From P & Q, infer Q.
c.BCF (Bold Conjunction Formation). From P and Q, infer P & Q.
5Show that the following are derivable axiom schemata in FLA:
a.FLD14. P → (P Q)
b.FLD15. Q → (P Q)
c.FLD16. (P Q) → (P Q)
d.FLD17. (P & Q) → (P Q)
6Prove that the following claim is true in FuzzyL: the value of the antecedent of the nth member of the infinite series of formulas (¬P → P) → Q,
(¬P → (¬P → P)) → Q, (¬P → (¬P → (¬P → P))) → Q, (¬P → (¬P → (¬P → (¬P → P)))) → Q, . . . , is the minimum of 1 and (n + 1) times the value of P. Hint: Show that the value of the antecedent of each formula after the first one is a particular function of the value of the antecedent of the previous formula.
7a. Show that R is semantically entailed by P ((P → Q) (P → (Q → R))) in FuzzyL.
b.Show that R is derivable from P ((P → Q) (P → (Q → R))) in the axiomatic system FLA.
c.Show that (P ((P → Q) (P → (Q → R)))) → R is not a tautology of FuzzyL.
SECTION 13.2
8Prove that
a.if m and n are rational values in the unit interval so is max (1, 1−m + n).
b.if m is a rational value in the unit interval then so is 1−m.
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Derivation Systems for Fuzzy Propositional Logic |
9a. Show in FLPA that the instance [1, 1] of FLP7 is derivable from the other axioms.
b. Show that all instances of FLP7 are derivable from the other axioms in FLPA.
10Derive [A ¬A, 1/2] in FLPA. (Hint: Use the corresponding derivation in Chapter 6 as a guide. Before using any derived rules or axioms from Chapter 6, be sure to establish that they are also derivable in FLPA.)
11Show that the rule
HS. From [P → Q, m] and [Q → R, n] infer [P → R, p] where p = max (0, m + n−1)
is derivable in FLPA.
12Derive graded versions of the following rules for FLPA (where the graded value of the inferred formula should be the least value that it can have given the least values of the formulas from which it is derived):
a.TRAN
b.GHS
c.CON
d.MT
13Show that the following are derivable as rules in FLPA:
a.BDF (Bold Disjunction Formation). From [P, m] and [Q, n] infer [P Q, p] where p = min (1, m + n)
b.WCF (Weak Conjunction Formation). From [P, m] and [Q, n] infer
[P Q, p] where p = min (m, n)
c.WCI (Weak Conjunction Inference). From [P Q, n] infer either of [P, n] or [Q, n]
d.DS (Disjunctive Syllogism). From [P Q, m] and [¬P, n] infer [Q, p] and from [P Q, m] and [¬Q, n] infer [P, p]
where p = max (0, m + n−1)
14Show that the rule
VS (Value Summary). From [m → ¬n, 1], [p → (q →m), 1] and [¬r →p, 1], infer [¬(q → ¬n) →r, 1]
is derivable in FLPA.
15 Consider the derived rule
DC. From [P → R, 1] and [Q → R, 1] infer [(P Q) → R, 1].
a.Modify the derivation of this rule in Section 13.2 so that it begins with graded formulas [P → R, m] and [Q → R, n], showing the correct graded values for each subsequent formula in the derivation.
b.The graded value obtained for (P Q) → R in the derivation in part (a) is too weak. Prove this by showing (semantically) that the least graded value that (P Q) → R can have is min (m, n) and then giving an example of specific values m and n such that the graded value for (P Q) → R in your derivation in part (a) is less than min (m, n).
c.Produce a justification for the fully graded rule
FDC. From [P → R, m] and [Q → R, n] infer [(P Q) → R, min (m, n)].
13.8 Exercises |
259 |
Hint: Look at the derivation of MCD for L3PA in Section 6.2 of Chapter 6 for an idea about how to modify your derivation in part (a).
16Produce a derivation of [D, .7] from [A → .5, 1], [B → .5, 1], [(A → B) → .9, 1],
[C → B, 1], [D → .4, 1], and [(C → ¬D) → .9, 1] in FLPA.
17Explain why the greatest value that m can be when the formula (n → m) → 9/10 is true is 1/10 less than n (the rational value of the formula n).
18In Section 13.3 we claimed that corresponding to the entailment in RFuzzyL of Q from the set consisting of the formula ¬P → Q and the infinitely many formulas in the series (¬P → P) → Q, (¬P → (¬P → P)) → Q, (¬P → (¬P → (¬P → P))) → Q,
(¬P → (¬P → (¬P → (¬P → P)))) → Q, . . . there are n-entailments of Q from finite subsets of in RFuzzyL with n as close to 1 as you can be without actually getting there. Produce a series of entailments that support this claim.
SECTION 13.3
19a. Prove that for any x and y in a BL-algebra, x y ≤ x.
b.Prove that for any x, y and z in a BL-algebra, x ≤ z and y ≤ z if and only if
xy ≤ z.
c.Prove that for any x, y, z, and w in a BL-algebra, if x ≤ z and y ≤ w, then
xy ≤ z w.
d.Prove that for any x, y and z in a BL-algebra, x (y z) = (x y) (x z).
20Show that each of the following axioms, when interpreted algebraically, evaluates to unit in every BL-algebra:
a.BL3
b.BL4
c.BL6
d.BL8.
21Show that the algebraic interpretation of Modus Ponens,
If unit ≤ x and unit ≤ x y, then unit ≤ y, is true in every BL-algebra.
22 Show that the algebraic interpretation of the axiom schema
BLL9. ¬¬P → P
holds true in every MV-algebra.
SECTION 13.4
23 Show that the algebraic interpretation of the axiom schema
BLG9. P → (P & P)
is true in every Godel¨ algebra.
24Prove that every set that contains the formula ¬GP →G Q and at least one formula from the infinite series
(¬GP →G P) →G Q
(¬GP →G (¬GP →G P)) →G Q
(¬GP →G (¬GP →G (¬GP →G P))) →G Q
(¬GP →G (¬GP → (¬GP →G (¬GP →G P)))) →G Q
. . .
entails Q in FuzzyG.
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Derivation Systems for Fuzzy Propositional Logic |
25a. Derive Q from ¬GP →G Q and (¬GP →G P) →G Q in BLPA.
b.Derive Q from¬GP →G Q and (¬GP →G (¬GP →G (¬GP →G P))) →G Q in BLPA.
26Prove Result 11.4, the Deduction Theorem for BLGA. Hint: Review the proof in Chapter 2 of the Deduction Theorem for classical propositional logic (Result 2.5, Section 2.4).
27Produce an example that shows that the FuzzyG conditional is not continuous.
SECTION 13.5
28 Show that the algebraic interpretations of the axiom schemata
BLP9. ¬P¬PP →P (((Q &P P) →P (R &P P)) →P (Q →P R))
and
BLP10. (P &P ¬PP) →P 0
are true in every product algebra.
29Derive the formula ¬P(P &P ¬PP) (without any assumptions) in BLPA.
30Show that the set * consisting of the formulas ¬P¬PR, (P P R) →P R)→P (Q P R), and the infinitely many formulas in the series
(((P P R) →P R) →P (P P R)) →P (Q P R)
(((P P R) →P R) →P (((P P R) →P R) →P (P P R))) →P (Q P R)
(((P P R) →P R) →P (((P P R) →P R) →P (((P P R) →P R) →P (P P R)))) →P (Q P R)
(((P P R) →P R) →P (((P P R) →P R) →P
(((P P R) →P R) →P (((P P R) →P R) →P (P P R))))) →P (Q P R)
. . .
semantically entails the formula Q P R in FuzzyP but that none of its finite subsets does.
To do this:
a.Assume that each of the formulas in * is true, and for each of the conditionals in the set explain what the value of R must be in order for the conditional’s antecedent to be true.
b.Noting that the value of R cannot be 0, because of the inclusion of ¬P¬PR in*, explain why the antecedents of one of the conditionals in the set must be true—for it will follow that the consequent of that conditional, Q P R, will also be true and hence the entailment from the set * holds.
c.For each formula S in *, show that if S is excluded from a finite subset ψ of * then ψ does not entail Q P R; that is, all of the formulas in ψ can be
true while Q P R is not.
31Prove Result 13.5, the Modified Deduction Theorem for BLPA. Hint: Review the proof of Result 13.3 in Section 13.1.
32Produce an example that shows that the FuzzyP conditional is not continuous.
13.8 Exercises |
261 |
SECTION 13.7
33Produce a derivation in FL PA that shows that if the formula P has the value .3 in FuzzyL, then ¬ P ¬ ¬P has the value 1.
34Produce a derivation in FL PA that shows that if the formula P has the value .5 in FuzzyL, then P → P also has the value .5.
35Produce a derivation that shows that the formula P ¬¬P is a theorem in BLG A.
36Produce a derivation that shows that the formula ( P & ¬P) → 0 is a theorem in BLP A.