=Paper=
{{Paper
|id=Vol-1494/paper6
|storemode=property
|title=Linguistic Logics with Hedges
|pdfUrl=https://ceur-ws.org/Vol-1494/paper6.pdf
|volume=Vol-1494
|dblpUrl=https://dblp.org/rec/conf/iwost/LeT15
}}
==Linguistic Logics with Hedges==
https://ceur-ws.org/Vol-1494/paper6.pdf
Linguistic Logics with Hedges
Van-Hung Le1 and Dinh-Khang Tran2
1
Faculty of Information Technology
Hanoi University of Mining and Geology, Vietnam
levanhung@humg.edu.vn
2
School of Information and Communication Technology
Hanoi University of Science and Technology, Vietnam
khangtd@soict.hust.edu.vn
Abstract. Human knowledge is commonly expressed linguistically, i.e.,
truth of vague sentences is given in linguistic terms, and many hedges
are often used simultaneously to state different levels of emphasis. In this
paper, we propose an axiomatization of mathematical fuzzy logic with
many hedges where each hedge does not have any dual one. Then, we
present linguistic logics built based on the proposed axiomatization and
the one in a previous work in order to make it easier to represent and
reason with linguistically-expressed human knowledge.
1 Introduction
Extending logical systems of mathematical fuzzy logic (MFL) with hedges is ax-
iomatized by Hájek [1], Vychodil [2], Esteva et al. [3], and Le et al. [4]. Hedges
are called truth-stressing or truth-depressing if they, respectively, strengthen or
weaken the meaning of the applied proposition. Intuitively, on a chain of truth
values, the truth function of a truth-depressing (resp., truth-stressing) hedge
(connective) is a superdiagonal (resp., subdiagonal) non-decreasing function pre-
serving 0 and 1. In [1–3], logical systems of MFL are extended by a truth-stressing
hedge and/or a truth-depressing one. However, in the real world, we often use
many hedges, e.g., very, rather, and slightly, simultaneously to express different
levels of emphasis. Moreover, human knowledge is commonly expressed linguis-
tically, i.e., truth of vague sentences is given in linguistic terms. Therefore, it is
possibly worth extending MFL with many truth-stressing and truth-depressing
hedges and using a linguistic truth domain (LTD) in order to make it easier to
represent and reason with linguistically-expressed human knowledge.
In this paper, first, we propose a new axiomatization for MFL with many
hedges which is simpler, but more general than the one in [4] since in this ax-
iomatization, hedges are not required to be pairwise dual. Then, we present
linguistic logics built based on the two axiomatizations. The linguistic logics use
an LTD having values such as true, very true and very slightly false. Thus, do-
mains and codomains of the hedge functions are the LTD. Since Gödel logic (G)
and Lukasiewicz logic (L) are two of the three basic t-norm based ones (the other
is product logic (Π)) which have received increasing attention in the last fifteen
years, we also define Gödel and Lukasiewicz operations on an LTD. Therefore,
we can have particular linguistic logics based on G or L.
The remainder of the paper is organized as follows. Section 2 gives an overview
of the notions and results of MFL. Section 3 presents LTDs, operations of G-
algebras and MV-algebras, and truth functions of hedges. Section 4 proposes an
axiomatization for many hedges where each hedge does not have any dual one.
Section 5 presents linguistic logics built based on the proposed axiomatization
and the one in [4]. Section 5 concludes the paper and outlines our future work.
2 Preliminaries on Mathematical Fuzzy Logic
Let L be a propositional logic in a language L, a set of connectives with finite
arity. A truth constant r is a special formula whose truth value under every
evaluation is r. Formulae are built from variables and truth constants using
connectives in L. Each evaluation e of variables by truth values uniquely extends
to an evaluation e(ϕ) of all formulae ϕ using truth functions of connectives. A
formula ϕ is called an 1-tautology if e(ϕ) = 1 for all evaluations e. Axioms of
the logic are taken from 1-tautology formulae. A theory is a set of formulae. An
evaluation e is called a model of a theory T if e(ϕ) = 1, ∀ϕ ∈ T . A proof in T is
a sequence ϕ1 , . . . , ϕn of formulae whose each member is either an axiom of the
logic or a member of T or follows from some preceding members of the sequence
using the deduction rule(s) of the logic. A formula ϕ is called provable, denoted
T `L ϕ, if ϕ is the last member of a proof in T . If T = ∅, ϕ is said to be provable
in the logic [5, 6]. L is a Rasiowa-implicative logic if there is a binary connective
→ satisfying reflexivity, weakening, and the following [6]:
(MP) ϕ, ϕ → ψ `L ψ
(T) ϕ → ψ, ψ → χ `L ϕ → χ
(sCng) ϕ → ψ, ψ → ϕ `L c(χ1 , . . . χi , ϕ, . . . , χn ) → c(χ1 , . . . χi , ψ, . . . , χn ),
for each n-ary c ∈ L and each i < n.
Every finitary Rasiowa-implicative logic L is algebraizable. Its equivalent alge-
braic semantics, a class of L-algebras, denoted L, is a quasivariety. The algebraic
semantics enjoys the following strong completeness [6].
Theorem 1. For every set Γ ∪ {ϕ} of formulae, Γ `L ϕ iff for every A ∈ L
and every A-model e of Γ , we have e(ϕ) = 1.
Each L-algebra A is endowed with a relation ≤ (called preorder ) by setting,
∀a, b ∈ A, a ≤ b iff a ⇒ b = 1, where ⇒ is the truth function of →. If ≤ is a
total order, A is called an L-chain. L is called a semilinear logic iff it is strongly
complete w.r.t. the class of L-chains [6]. Most logical systems called fuzzy logics
are a finitary Rasiowa-implicative semilinear logic. They belong to a large class
of systems that are axiomatic expansions of MTL satisfying (sCng) for any new
connective [6]. The systems are called core fuzzy logics. Well-known examples
are basic logic (BL), G, L, Π [5], SBL, NM, MTL, and SMTL [6].
Definition 1. [6] Let L be a core fuzzy logic and K a class of L-chains. L has
the (finite) strong K-completeness property, (F)SKC, if for every (finite) set of
formulae Γ and every formula ϕ, it holds that Γ `L ϕ iff e(ϕ) = 1 for every
L-algebra A ∈ K and each A-model e of Γ . L has the K-completeness property,
KC, when the equivalence is true for Γ = ∅.
When K is the class of all chains whose support is the unit interval [0, 1] with
the usual ordering, the (F)SKC can be called the (finite) strong standard com-
pleteness, (F)SSC.
Theorem 2. [6] Let L be a core fuzzy logic and K a class of L-chains. Then, (i)
L has the SKC iff every countable L-chain is embeddable into some member of K,
and (ii) if the language of L is finite, L has the FSKC iff every countable L-chain
is partially embeddable into some member of K, i.e., for every finite partial of a
countable L-chain, there is a one-to-one mapping preserving the operations into
some member of K.
The language of BL [5, 6] consists of the primitive connectives &, → and the truth
constant 0. Further definable connectives are as follows: ϕ ∧ ψ ≡ ϕ&(ϕ → ψ),
ϕ ∨ ψ ≡ ((ϕ → ψ) → ψ) ∧ ((ψ → ϕ) → ϕ), ϕ ↔ ψ ≡ (ϕ → ψ)&(ψ → ϕ),
¬ϕ ≡ ϕ → 0, and 1 ≡ ¬0. Axioms of BL [6] are the following:
(BL1) (ϕ → ψ) → ((ψ → χ) → (ϕ → χ))
(BL4) ϕ&(ϕ → ψ) → ψ&(ψ → ϕ)
(BL5a) (ϕ&ψ → χ) → (ϕ → (ψ → χ))
(BL5b) (ϕ → (ψ → χ)) → (ϕ&ψ → χ)
(BL6) ((ϕ → ψ) → χ) → (((ψ → ϕ) → χ) → χ)
(BL7) 0 → ϕ
The only deduction rule of BL is modus ponens (MP).
The axiom system of G (resp., L) is an extension of that of BL by the axiom:
(G) ϕ → ϕ&ϕ (resp., (L) ¬¬ϕ → ϕ). G proves that ϕ&ψ is ϕ ∧ ψ. The following
formulae are provable in MTL, BL, G, and L[6]:
(ϕ → ψ) → (¬ψ → ¬ϕ), (1)
(ϕ → ψ) ∨ (ψ → ϕ). (2)
Let ∗, ⇒, ∩, ∪, − denote truth functions of connectives &, →, ∧, ∨, ¬, respectively.
For BL, G and L, the truth functions ∩ and ∪ are min and max, respectively.
A residuated lattice is an algebra hA, ∗, ⇒, ∩, ∪, 0, 1i of type h2, 2, 2, 2, 0, 0i
such that: (i) hA, ∩, ∪, 0, 1i is a lattice with the largest element 1 and least
element 0 (w.r.t. the lattice ordering ≤); (ii) hA, ∗, 1i is a commutative semigroup
with the unit element 1, i.e., * is commutative, associative, 1∗x = x for all x; (iii)
⇒ is the residuum of ∗, i.e., for each x, y, z ∈ A holds: x ∗ y ≤ z iff x ≤ y ⇒ z.
A residuated lattice hA, ∗, ⇒, ∩, ∪, 0, 1i is a BL-algebra iff the following iden-
tities hold for all x, y ∈ A: (i) Prelinear : (x ⇒ y) ∪ (y ⇒ x) = 1; and (ii)
Divisible: x ∩ y = x ∗ (x ⇒ y). BL-algebras satisfying x ∗ x = x are called G-
algebras, while BL-algebras satisfying − − x = x are called MV-algebras. The
class of BL-algebras, G-algebras, and MV-algebras are the equivalent algebraic
semantics of BL, G, and L, respectively.
3 Linguistic Truth Domains and Operations
3.1 Linguistic Truth Domains
Let V, H, R, S, T, and F stand for Very, Highly, Rather, Slightly, True, and
False, respectively. In hedge algebra (HA) theory [7, 8], values of the linguistic
variable Truth, e.g., VT and VSF, can be regarded as being generated from a set
of primary terms G = {F, T } using hedges from a set H = {V, S, . . . } as unary
operations. There exists a natural ordering among them, e.g., ST < T. Thus, a
term domain of Truth is a partially ordered set (poset) and can be characterised
by an HA X = (X , G, H, ≤), where X is a term set, G is a set of primary terms,
H is a set of hedges, and ≤ is a semantic order relation (SOR) on X .
There are natural semantic properties of hedges and terms. Hedges either
increase or decrease the meaning of terms, i.e., ∀h ∈ H, ∀x ∈ X , either hx ≥
x or hx ≤ x. It is denoted by h ≥ k if a hedge h modifies terms more than or
equal to another hedge k, i.e., ∀x ∈ X , hx ≤ kx ≤ x or x ≤ kx ≤ hx. Since H
and X are disjoint, the same notation ≤ can be used for different order relations
on H and X without confusion. A hedge has a semantic effect on others. If
h strengthens the degree of modification of k, i.e., ∀x ∈ X , hkx ≤ kx ≤ x or
x ≤ kx ≤ hkx, then h is positive w.r.t. k. If h weakens the degree of modification
of k, i.e., ∀x ∈ X , kx ≤ hkx ≤ x or x ≤ hkx ≤ kx, then h is negative w.r.t. k. An
important semantic property of hedges, called semantic heredity, is that hedges
change the meaning of a term, but somewhat preserve its original meaning. Thus,
if hx ≤ kx, where x ∈ X , then H(hx) ≤ H(kx), where H(u) denotes the set of
all terms generated from u by means of hedges, i.e., H(u) = {σu|σ ∈ H∗ }, where
H∗ is the set of all strings of symbols in H including the empty one.
For Truth, the primary terms F ≤ T are denoted by c− and c+ , respectively.
H can be divided into two disjoint subsets H+ and H− by H+ = {h|hc+ >
c+ } = {h|hc− < c− } and H− = {h|hc+ < c+ } = {h|hc− > c− }. For example,
H = {V, H, R, S} is decomposed into H+ = {V, H} and H− = {R, S}. Hedges
in each of H+ and H− may be comparable. So, H+ and H− become posets. Let
I∈ / H be an artificial hedge, called the identity, defined by ∀x ∈ X , Ix = x. I is
the least element in each of H+ ∪ {I} and H− ∪ {I}. An HA is said to be linear
if both H+ and H− are linearly ordered. The term domain X of a linear HA is
also linearly ordered. In this paper, we restrict ourselves to linear HAs.
Example 1. The HA X = (X , {c− , c+ }, H = {V, H, R, S}, ≤) is a linear HA
since in H+ , we have H < V , and in H− , we have R < S.
A linguistic truth domain (LTD) X taken from a linear HA X = (X , {c− , c+ }, H, ≤
) is the linearly ordered set X = X ∪{0, W, 1}, where 0 (AbsolutelyFalse), W (the
middle truth value), and 1 (AbsolutelyTrue) are the least, neutral and greatest
elements of X, respectively [9], and ∀x ∈ {0, W, 1} and ∀h ∈ H, hx = x.
An extended order relation ≤e on H ∪ {I} is defined as follows [9]: ∀h, k ∈
H ∪ {I}, h ≤e k if: (i) h ∈ H− , k ∈ H+ ; or (ii) h, k ∈ H+ ∪ {I} and h ≤ k; or
(iii) h, k ∈ H− ∪ {I} and h ≥ k. It is denoted by h 0 vn if i ≤ j
vi ∗ vj = , vi ⇒ vj = .
v0 otherwise vn+j−i otherwise
The negation is defined by: given x = σc, where σ ∈ H∗ and c ∈ {c+ , c− }, we
have y = −x, if y = σc0 and {c, c0 } = {c+ , c− }, e.g., V c+ = −V c− , V c− = −V c+ .
3.4 Truth Functions of Hedge Connectives
Definition 2. [10] Let X = (X , {c+ , c− }, H, ≤) be a linear HA. Truth functions
h• : X → X of all hedges h ∈ H ∪ {I} satisfy the following conditions:
for all x ∈ {0, W, 1}, h• (x) = x (3)
•
for all x ∈ X, I (x) = x (4)
h• (hc+ ) = c+ (5)
if x ≥ y, h• (x) ≥ h• (y) (6)
for all k ∈ H ∪ {I} such that h ≤e k, h• (x) ≥ k • (x) (7)
By (7), given the hedges in Example 1, we have ∀x ∈ X, V • (x) ≤ H • (x) ≤
x ≤ R• (x) ≤ S • (x). This is in accordance with fuzzy-set-based interpreta-
tions of hedges [11], which satisfy the semantic entailment [12]: x is veryA ⇒
x is highlyA ⇒ x is A ⇒ x is rather A ⇒ x is slightlyA, where A is a fuzzy
predicate. Truth functions of hedges always exist [9].
Example 2. Consider a 2-limit HA X = (X , {c+ , c− }, {V, H, R, S}, ≤). Table 1
gives an example of truth functions of hedges, where the value of a truth function,
in the first row, of a value x, in the first column, is in the corresponding cell, e.g.,
V • (V Rc+ ) = V Sc+ . Truth values in the first column are in ascending order.
Table 1. Truth functions of hedge connectives
V• H• R• S•
0 0 0 0 0
kV c− V V c− V V c− kHc− c− a
− − − −
kHc V V c kV c c kRc− a
c− V c− Hc− Rc− Sc−
V Rc− V Hc− RHc− SSc− V Sc−
HRc− HHc− SHc− RSc− V Sc−
Rc− Hc− c− Sc− V Sc−
RRc− RHc− V Rc− HSc− V Sc−
SRc− SHc− V Rc− V Sc− V Sc−
SSc− SHc− V Rc− V Sc− V Sc−
RSc− SHc− HRc− V Sc− V Sc−
Sc− c− Rc− V Sc− V Sc−
HSc− V Rc− RRc− V Sc− V Sc−
V Sc− RRc− SRc− V Sc− V Sc−
W W W W W
V Sc+ V Sc+ V Sc+ SRc+ RRc+
HSc+ V Sc+ V Sc+ RRc+ V Rc+
Sc+ V Sc+ V Sc+ Rc+ c+
RSc+ V Sc+ V Sc+ HRc+ SHc+
SSc+ V Sc+ V Sc+ V Rc+ SHc+
SRc+ V Sc+ V Sc+ V Rc+ SHc+
RRc+ V Sc+ HSc+ V Rc+ RHc+
Rc+ V Sc+ Sc+ c+ Hc+
HRc+ V Sc+ RSc+ SHc+ HHc+
V Rc+ V Sc+ SSc+ RHc+ V Hc+
c+ Sc+ Rc+ Hc+ V c+
kHc+ kRc+ c+ kV c+ V V c+ a
kV c+ c+ kHc+ V V c+ V V c+ a
1 1 1 1 1
a
k is any of the hedges, including the identity I.
4 An Axiomatization for Many Hedges
A hedge may modify truth more than another [11, 7, 8]. For example, S (resp.,
V ) modify truth more than R (resp., H ) since ST < RT < T (resp., T < HT
< VT ). To ease the presentation, let s0 , d0 denote the identity connective, i.e.,
for all ϕ, ϕ ≡ s0 ϕ ≡ d0 ϕ, and their truth functions s•0 and d•0 are the identity.
Definition 3. Let L be a core fuzzy logic. A logic Lp,q s,d , where p, q are positive
integers, is an expansion of L with new unary connectives s1 , ..., sp (for truth-
stressers) and d1 , ..., dq (for truth-depressers) by the following additional axioms,
for i = 1, ..., p and j = 1, ..., q:
(Si ) si ϕ → si−1 ϕ (Sp+1 ) sp 1 (Dj ) dj−1 ϕ → dj ϕ (Dq+1 ) ¬dq 0
and the following additional deduction rule:
(DRh ) from (ϕ → ψ) ∨ χ infer (hϕ → hψ) ∨ χ, for each h ∈ {s1 , ..., sp , d1 , ..., dq }.
Axiom (Si ) (resp., axiom (Dj )) expresses that si (resp., dj ) modifies truth more
than si−1 (resp., dj−1 ), for i = 2, ..., p (resp., j = 2, ..., q).
Lemma 1. The following deductions are valid, for i = 1, ..., p and j = 1, ..., q:
`Lp,q
s,d
si ϕ → ϕ (8)
`Lp,q
s,d
ϕ → dj ϕ (9)
`Lp,q
s,d
¬si 0 (10)
`Lp,q
s,d
si 1 (11)
`Lp,q
s,d
dj 1 (12)
`Lp,q
s,d
¬dj 0 (13)
ϕ → ψ `Lp,q
s,d
si ϕ → si ψ (14)
ψ `Lp,q
s,d
si ψ (15)
ϕ → ψ `Lp,q
s,d
dj ϕ → dj ψ (16)
dj ϕ, ϕ → ψ ` Lp,q
s,d
dj ψ (17)
si ϕ, ϕ → ψ `Lp,q
s,d
si ψ (18)
si ϕ, si (ϕ → ψ) `Lp,q
s,d
si ψ (19)
Proof. (8) follows from (Si ),...,(S1 ), and (T).
(9) follows from (D1 ),...,(Dj ), and (T).
(10) follows immediately from (8) taking ϕ = 0.
(11) `Lp,q
s,d
sp ϕ → si ϕ (for i < p, by (Sp ), ..., (Si+1 ) and (T)),
`Lp,q
s,d
sp 1 → si 1 (by taking ϕ = 1), `Lp,q
s,d
si 1 (by (Sp+1 ) and (MP)).
(12) follows immediately from (9) taking ϕ = 1.
(13) `Lp,q
s,d
dj ϕ → dq ϕ (for j < q, by (Dj+1 ), ..., (Dq ), and (T)),
`Ls,d ¬dq ϕ → ¬dj ϕ (by (1) and (MP)),
p,q
`Lp,q
s,d
¬dq 0 → ¬dj 0 (by taking ϕ = 0), `Lp,q s,d
¬dj 0 (by (Dq+1 ) and (MP)).
(14) follows directly from Rule (DRh ) taking χ = 0, h = si .
(15) follows from (14) taking ϕ = 1 and using (11).
(16) follows directly from Rule (DRh ) taking χ = 0, h = dj .
(17) follows immediately from (16) and (MP).
(18) follows immediately from (14) and (MP).
(19) follows from (8), (MP) and (18).
Properties (10)-(13) imply that truth functions of si and dj preserve 0 and
1. Property (15) is the necessitation deduction rule of Hájek’s and Vychodil’s
axiomatizations [1, 2]. Properties (17) and (18) are a stronger version of modus
ponens: if ϕ implies ψ, then very (resp., slightly) ϕ implies very (resp., slightly)
ψ. Property (19) is a deduction version of the K-like axiom used in the previous
axiomatizations of logics with hedges [1, 2]. Since (14) and (16) express that
Property (sCng) is satisfied for si and dj , Lp,q
s,d is a finitary Rasiowa-implicative
logic, and its equivalent algebraic semantics is the class of Lp,q
s,d -algebras.
Definition 4. An algebra A=hA, ∗, ⇒, ∩, ∪, 0, 1, s•1 , ..., s•p , d•1 , ..., d•q i of type
h2, 2, 2, 2, 0, 0, 1, ..., 1i is an Lp,q
s,d -algebra if it is an L-algebra expanded by unary
• •
operators si , dj : A → A that satisfy, for all x, y, z ∈ A, i = 1, p and j = 1, q,
s•i (x) ≤ s•i−1 (x) (20)
s•p (1) = 1 (21)
d•j (x) ≥ d•j−1 (x) (22)
d•q (0) = 0 (23)
if (x ⇒ y) ∪ z = 1 then (s•i (x) ⇒ s•i (y)) ∪ z = 1 (24)
if (x ⇒ y) ∪ z = 1 then (d•j (x) ⇒ d•j (y)) ∪ z = 1 (25)
where s•i and d•j are truth functions of connectives si and dj , for all i = 1, ..., p
and j = 1, ..., q, respectively.
By (20), s•i (x) ≤ x, i.e., s•i is subdiagonal, for all i = 1, p. By (22), d•j is superdiag-
onal, for all j = 1, q. In a chain of truth values, the quasiequations (24) and (25)
turn out to be equivalently expressed by: if x ⇒ y = 1, then s•i (x) ⇒ s•i (y) = 1
and d•j (x) ⇒ d•j (y) = 1, respectively, i.e., s•i and d•j are non-decreasing.
If hA, ∗, ⇒, ∩, ∪, 0, 1i is an L-chain, and s•i and d•j satisfy (20)-(25), the ex-
panded structure A, ∗, ⇒, ∩, ∪, 0, 1, s•1 , ..., s•p , d•1 , ..., d•q is an Lp,q
s,d -chain.
Theorem 3. Let L be a core fuzzy logic, K a class of L-chains, and Kp,q s,d the
class of the Lp,q
s,d -chains whose s1 , ..., s ,
p 1d , ..., d q -free reducts are in K. Then: (i)
p,q p,q
Ls,d is a conservative expansion of L; (ii) Ls,d is strongly complete w.r.t. the
class of all Lp,q p,q
s,d -chains, i.e., Ls,d is semilinear; (iii) L has the FSSC, FSKC,
p,q
SSC, and SKC iff Ls,d has the FSSC, FSKC, SSC, and SKC, respectively.
Proof. (i) Let L be the language of L. We show that, for every set Γ ∪ {ϕ}
of L-formulae, Γ `Lp,q s,d
ϕ iff Γ `L ϕ. Obviously, if Γ `L ϕ then Γ `Lp,q s,d
ϕ. If
Γ 6`L ϕ, there is an L-chain A and an A-evaluation e such that e is A-model of
Γ and e(ϕ) 6= 1. A can be expanded to an Lp,q s,d -chain A’ by defining si (1) = 1;
∀a ∈ A \ {1}, si (a) = 0; dj (0) = 0; and ∀a ∈ A \ {0}, dj (a) = 1, for all i = 1, p
and j = 1, q. Thus, in the expanded language, we have Γ 6`Lp,q s,d
ϕ.
p,q p,q
(ii) Since ∨ remains a disjunction in Ls,d and (2) is valid in L, Ls,d is semilinear.
(iii) We prove for the case of the SSC, and the others can be done analogously.
Since Lp,q p,q
s,d is a conservative expansion of L, if Ls,d has the SSC, so does L. As-
sume that L has the SSC. We show that any countable Lp,q s,d -chain A can be
embedded into a standard Lp,q s,d -chain. By Theorem 2, the s1 , ..., sp , d1 , ..., dq -free
reduct of A can be embedded into a standard L-chain B = h[0, 1], ∗, ⇒, ∩, ∪, 0, 1i
by a mapping f . Since A is countable, for each 1 ≤ k ≤ p, we may arrange
all points {hf (x), f (sk (x))i|x ∈ A} into a sequence {hf (xi ), f (sk (xi ))i|xi ∈
A, i = 1, 2, . . . }, where 0 = x1 < x2 < . . . and limi→∞ xi = 1. Let s0k :
[0, 1] → [0, 1] be the piecewise linear function connecting neighboured points
from {hf (xi ), f (sk (xi ))i}. Similarly, for each 1 ≤ l ≤ q, let d0l be the piecewise
linear function connecting neighboured points from {hf (xi ), f (dl (xi ))i}. It can
be shown that all s0k and d0l satisfy (20)-(25). Hence, B expanded by all s0k and
d0l is a standard Lp,qs,d -chain into which A is embedded.
5 Linguistic Logics with Hedges
5.1 Linguistic Logic with Many Hedges
Let L be a core fuzzy logic. Given a linear HA, we can build a linguistic logic with
many hedges based on L and the HA. For instance, given the HA in Example
1, X = (X , {c− , c+ }, H = {V, H, R, S}, ≤), a linguistic logic, denoted Llh , is an
expansion of L with new unary connectives V, H, R, S by the following axioms:
(S1lh ) Hϕ → ϕ (S2lh ) V ϕ → Hϕ (S3lh ) V1
(D1lh ) ϕ → Rϕ (D2lh ) Rϕ → Sϕ (D3lh ) ¬S0
and the following additional deduction rule:
(DRlh ) from (ϕ → ψ) ∨ χ infer (hϕ → hψ) ∨ χ, for each h ∈ {V, H, R, S}.
The equivalent algebraic semantics of Llh is the class of Llh -algebras, denoted
Llh . Llh -algebras utilize a linear linguistic domain X taken from the HA.
An Llh -algebra is an L-algebra expanded by unary non-decreasing operators
V • , H • , R• , S • : X → X satisfying, for all x ∈ X,
H • (x) ≤ x, V • (x) ≤ H • (x), V • (1) = 1,
R• (x) ≥ x, S • (x) ≥ R• (x), S • (0) = 0.
Theorem 4 (Strong Completeness). For every set Γ ∪ {ϕ} of formulae,
Γ `Llh ϕ iff for every A ∈ Llh and every A-model e of Γ , e(ϕ) = 1.
In particular, given the Gödel and Lukasiewicz operations respectively defined
in Subsections 3.2 and 3.3 and truth functions of hedges in Example 2, we can
have linguistic logics based on G or L with the well-defined operators.
5.2 Mathematical Fuzzy logic with Many Dual Hedges
It can be observed that each hedge can have a dual one, e.g., slightly and rather
can be seen as a dual hedge of very and highly, respectively. Thus, there might
be axioms expressing dual relations of hedges in addition to axioms expressing
their comparative truth modification strength.
Definition 5. [4] Let L be a core fuzzy logic. A logic L2n s,d , where n is a positive
integer, is an expansion of L with new unary connectives s1 , ..., sn (for truth-
stressers) and d1 , ..., dn (for truth-depressers) by the following additional axioms,
for i = 1, ..., n:
(Sidh ) si ϕ → si−1 ϕ dh
(Sn+1 ) sn 1 (Didh ) di−1 ϕ → di ϕ (SDidh ) di ϕ → ¬si ¬ϕ
and the following additional deduction rule:
(DRdh ) from (ϕ → ψ) ∨ χ infer (hϕ → hψ) ∨ χ, for h ∈ {s1 , ..., sn , d1 , ..., dn }.
The logic L2ns,d is L expanded by 2n hedges, where hedges are divided into pairs
of dual ones. Axiom (SDi ) expresses the dual relation between hedges si and di
and coincides with Axiom (ST2) in Vychodil’s axiomatization. For the case of
very, slightly, and ϕ = young, it means “slightly young implies not very old ”.
L2n
s,d is also a finitary Rasiowa-implicative logic, and its equivalent algebraic
semantics is the class of L2ns,d -algebras.
Definition 6. [4] An algebra A=hA, ∗, ⇒, ∩, ∪, 0, 1, s•1 , ..., s•n , d•1 , ..., d•n i of type
h2, 2, 2, 2, 0, 0, 1, ..., 1i is an L2n
s,d -algebra if it is an L-algebra expanded by unary
operators s•i , d•i : A → A that satisfy, for all x, y, z ∈ A and i = 1, ..., n,
s•i (x) ≤ s•i−1 (x), s•n (1) = 1,
d•i (x) ≥ d•i−1 (x), di (x) ≤ −s•i (−x),
•
if (x ⇒ y) ∪ z = 1 then (s•i (x) ⇒ s•i (y)) ∪ z = 1,
if (x ⇒ y) ∪ z = 1 then (d•i (x) ⇒ d•i (y)) ∪ z = 1.
where s•i and d•i are truth functions of connectives si and di , respectively.
Theorem 5. [4] Let L be a core fuzzy logic, K a class of L-chains, and K2n s,d the
class of the L2ns,d -chains whose s 1 , ..., s ,
n 1d , ..., d n -free reducts are in K. Then:
2n 2n
(i) Ls,d is a conservative expansion of L; (ii) Ls,d is strongly complete w.r.t. the
class of all L2n 2n
s,d -chains, i.e., Ls,d is semilinear; (iii) L has the FSSC, FSKC,
SSC, and SKC iff L2n s,d has the the FSSC, FSKC, SSC, and SKC, respectively.
It can be seen that in a case when there is one truth-stressing (resp., truth-
depressing) hedge without a dual one, we just add the axioms expressing its
relations to the existing truth-stressing (resp., truth-depressing) hedges accord-
ing to their comparative truth modification strength.
5.3 Linguistic Logic with Many Dual Hedges
Let L be a core fuzzy logic. Given a linear HA, we can build a linguistic logic
with many dual hedges based on L. For example, given the HA in Example 1,
X = (X , {c− , c+ }, H = {V, H, R, S}, ≤), a linguistic logic, denoted Lldh , is an
expansion of L with new unary connectives V, H, R, S by the following axioms:
(S1ldh ) Hϕ → ϕ (S2ldh ) V ϕ → Hϕ (S3ldh ) V 1 (D1ldh ) ϕ → Rϕ
(D2ldh ) Rϕ → Sϕ (SD1ldh ) Rϕ → ¬H¬ϕ (SD2ldh ) Sϕ → ¬V ¬ϕ
and the following additional deduction rule:
(DRldh ) from (ϕ → ψ) ∨ χ infer (hϕ → hψ) ∨ χ, for each h ∈ {V, H, R, S}.
The equivalent algebraic semantics of Lldh is the class of Lldh -algebras, denoted
Lldh . Lldh -algebras utilize a linear linguistic domain X taken from the HA.
An Lldh -algebra is an L-algebra expanded by unary non-decreasing operators
V , H • , R• , S • : X → X satisfying, for all x ∈ X,
•
H • (x) ≤ x, V • (x) ≤ H • (x), V • (1) = 1, R• (x) ≥ x,
• • • • • •
S (x) ≥ R (x), S (x) ≤ −V (−x), R (x) ≤ −H (−x).
Taking into account that ∀x ∈ X, R• (x) = −H • (−x), S • (x) = −V • (−x), we
can see that truth functions of hedges in Example 2 satisfy the above conditions.
Theorem 6 (Strong Completeness). For every set Γ ∪ {ϕ} of formulae,
Γ `Lldh ϕ iff for every A ∈ Lldh and every A-model e of Γ , e(ϕ) = 1.
6 Acknowledgments
This research is funded by Vietnam National Foundation for Science and Tech-
nology Development (NAFOSTED) under grant number 102.04-2013.21.
7 Conclusion and Future Work
This paper proposes an axiomatization for mathematical fuzzy logic with many
hedges, where each hedge does not have any dual one. Then, based on the pro-
posed axiomatization and the one in a previous work, it proposes linguistic logics
for representing and reasoning with linguistically-expressed human knowledge,
where truth of vague sentences is given in linguistic terms, and many hedges
are often used simultaneously to express different levels of emphasis. For future
work, we will study first-order fuzzy logics with hedges.
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