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 di erent 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.
Extending logical systems of mathematical fuzzy logic (MFL) with hedges is
axiomatized by Hajek [
In this paper, rst, we propose a new axiomatization for MFL with many
hedges which is simpler, but more general than the one in [
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
Galgebras 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 [
Let L be a propositional logic in a language L, a set of connectives with nite
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(') of all formulae ' using truth functions of connectives. A
formula ' is called an 1-tautology if e(') = 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(') = 1, 8' 2 T . A proof in T is
a sequence '1; : : : ; 'n 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 ' is called provable, denoted
T `L ', if ' is the last member of a proof in T . If T = ;, ' is said to be provable
in the logic [
`L
' !
(sCng) ' ! ;
for each n-ary c 2 L and each i < n:
! `L ' !
! ' `L c( 1; : : : i; '; : : : ; n) ! c( 1; : : : i; ; : : : ; n);
Every nitary Rasiowa-implicative logic L is algebraizable. Its equivalent
algebraic semantics, a class of L-algebras, denoted L, is a quasivariety. The algebraic
semantics enjoys the following strong completeness [
Theorem 1. For every set [ f'g of formulae, and every A-model e of , we have e(') = 1. `L ' i for every A 2 L
Each L-algebra A is endowed with a relation (called preorder ) by setting,
8a; b 2 A, a b i 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 i it is strongly
complete w.r.t. the class of L-chains [
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 ( nite) strong standard completeness, (F)SSC.
Theorem 2. [
The language of BL [
) ! (( (BL4) '&(' ! ) ! (BL5a) ('& ! ) ! (' ! ( (BL5b) (' ! ( ! )) ! ('& (BL6) ((' ! ) ! ) ! ((( (BL7) 0 ! ' ! ) ! (' ! &( ! ') ! )) ! ) ! ') ! )) ) ! ) 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) ' ! '&' (resp., (L) ::' ! '). G proves that '& is ' ^ . The following
formulae are provable in MTL, BL, G, and L[
) ! (: (' ! ) _ ( ! :'); ! '): (1) (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 2 A holds: x y z i x y ) z.
A residuated lattice hA; ; ); \; [; 0; 1i is a BL-algebra i the following identities hold for all x; y 2 A: (i) Prelinear : (x ) y) [ (y ) x) = 1; and (ii) Divisible: x \ y = x (x ) y). BL-algebras satisfying x x = x are called Galgebras, 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 3.1
Let V, H, R, S, T, and F stand for Very, Highly, Rather, Slightly, True, and
False, respectively. In hedge algebra (HA) theory [
There are natural semantic properties of hedges and terms. Hedges either increase or decrease the meaning of terms, i.e., 8h 2 H; 8x 2 X ; either hx x or hx x. It is denoted by h k if a hedge h modi es terms more than or equal to another hedge k, i.e., 8x 2 X , hx kx x or x kx hx. Since H and X are disjoint, the same notation can be used for di erent order relations on H and X without confusion. A hedge has a semantic e ect on others. If h strengthens the degree of modi cation of k, i.e., 8x 2 X , hkx kx x or x kx hkx, then h is positive w.r.t. k. If h weakens the degree of modi cation of k, i.e., 8x 2 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 2 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) = f uj 2 H g, 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+ = fhjhc+ > c+g = fhjhc < c g and H = fhjhc+ < c+g = fhjhc > c g. For example, H = fV; H; R; Sg is decomposed into H+ = fV; Hg and H = fR; Sg. Hedges in each of H+ and H may be comparable. So, H+ and H become posets. Let I 2= H be an arti cial hedge, called the identity, de ned by 8x 2 X , Ix = x. I is the least element in each of H+ [ fIg and H [ fIg. 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 ; fc ; c+g; H = fV; H; R; Sg; ) 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 ; fc ; c+g; H;
) is the linearly ordered set X = X [f0; W; 1g, where 0 (AbsolutelyFalse), W (the
middle truth value), and 1 (AbsolutelyTrue) are the least, neutral and greatest
elements of X, respectively [
An extended order relation e on H [ fIg is de ned as follows [
Godel t-norm, its residuum, and negation can be de ned on an LTD X as: y1 iofthyerwxise ; x = 01 iofthxe=rw0ise : x y = min(x; y);
x ) y = 3.3
To have well-de ned operations, we consider only nitely many truth values.
An l-limit HA, where l is a positive integer, is a linear HA in which all terms
have a length of at most l + 1. An LTD taken from an l -limit HA is nite [
: and c 2 fc+; c g, we
V c ; V c = V c+. 3.4
De nition 2. [
e k; h (x) for all x 2 f0; W; 1g; h (x) = x for all x 2 X; I (x) = x
h (hc+) = c+
if x
y; h (x)
h (y)
k (x)
(3)
(4)
(5)
(6)
(7)
By (7), given the hedges in Example 1, we have 8x 2 X, V (x) H (x)
x R (x) S (x). This is in accordance with fuzzy-set-based
interpretations of hedges [
Example 2. Consider a 2-limit HA X = (X ; fc+; c g; fV; H; R; Sg; ). Table 1 gives an example of truth functions of hedges, where the value of a truth function, in the rst row, of a value x, in the rst column, is in the corresponding cell, e.g., V (V Rc+) = V Sc+. Truth values in the rst column are in ascending order. < VT ). To ease the presentation, let s0; d0 denote the identity connective, i.e., for all ', ' s0' d0', and their truth functions s0 and d0 are the identity. De nition 3. Let L be a core fuzzy logic. A logic Lsp;;dq, where p; q are positive integers, is an expansion of L with new unary connectives s1; :::; sp (for truthstressers) and d1; :::; dq (for truth-depressers) by the following additional axioms, for i = 1; :::; p and j = 1; :::; q: (Si) si' ! si 1' (Sp+1) sp1 (Dj ) dj 1' ! dj ' (Dq+1) :dq0 and the following additional deduction rule: (DRh) from (' ! ) _
infer (h' ! h ) _ ; for each h 2 fs1; :::; sp; d1; :::; dqg: Axiom (Si) (resp., axiom (Dj )) expresses that si (resp., dj ) modi es 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: `Lsp;;dq si' ! ' `Lsp;;dq ' ! dj ' `Lsp;;dq :si0 (8) (9) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) Proof. (8) follows from (Si),...,(S1), and (T). (9) follows from (D1),...,(Dj ), and (T). (10) follows immediately from (8) taking ' = 0. (11) `Lsp;;dq sp' ! si' (for i < p, by (Sp); :::; (Si+1) and (T)), `Lsp;;dq sp1 ! si1 (by taking ' = 1), `Lsp;;dq si1 (by (Sp+1) and (MP)). (12) follows immediately from (9) taking ' = 1. (13) `Lsp;;dq dj ' ! dq' (for j < q, by (Dj+1); :::; (Dq); and (T)), `Lsp;;dq :dq' ! :dj ' (by (1) and (MP)), `Lsp;;dq :dq0 ! :dj 0 (by taking ' = 0), `Lsp;;dq :dj 0 (by (Dq+1) and (MP)). (14) follows directly from Rule (DRh) taking = 0, h = si. (15) follows from (14) taking ' = 1 and using (11). (16) follows directly from Rule (DRh) taking = 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 Hajek's and Vychodil's
axiomatizations [
By (20), si (x) x, i.e., si is subdiagonal, for all i = 1; p. By (22), dj is superdiagonal, 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 si (x) ) si (y) = 1 and dj (x) ) dj (y) = 1, respectively, i.e., si and dj are non-decreasing.
If hA; ; ); \; [; 0; 1i is an L-chain, and si and dj satisfy (20)-(25), the expanded structure A; ; ); \; [; 0; 1; s1; :::; sp; d1; :::; dq is an Lsp;;dq-chain. Theorem 3. Let L be a core fuzzy logic, K a class of L-chains, and Ksp;;dq the class of the Lsp;;dq-chains whose s1; :::; sp; d1; :::; dq-free reducts are in K. Then: (i) Lsp;;dq is a conservative expansion of L; (ii) Lsp;;dq is strongly complete w.r.t. the class of all Lsp;;dq-chains, i.e., Lsp;;dq is semilinear; (iii) L has the FSSC, FSKC, SSC, and SKC i Lsp;;dq has the FSSC, FSKC, SSC, and SKC, respectively. Proof. (i) Let L be the language of L. We show that, for every set [ f'g of L-formulae, `Lsp;;dq ' i `L '. Obviously, if `L ' then `Lsp;;dq '. If 6`L ', there is an L-chain A and an A-evaluation e such that e is A-model of and e(') 6= 1. A can be expanded to an Lsp;;dq-chain A' by de ning si(1) = 1; 8a 2 A n f1g, si(a) = 0; dj(0) = 0; and 8a 2 A n f0g, dj(a) = 1, for all i = 1; p and j = 1; q. Thus, in the expanded language, we have 6`Lsp;;dq '. (ii) Since _ remains a disjunction in Lsp;;dq and (2) is valid in L, Lsp;;dq is semilinear. (iii) We prove for the case of the SSC, and the others can be done analogously. Since Lsp;;dq is a conservative expansion of L, if Lsp;;dq has the SSC, so does L. Assume that L has the SSC. We show that any countable Lsp;;dq-chain A can be embedded into a standard Lsp;;dq-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 fhf (x); f (sk(x))ijx 2 Ag into a sequence fhf (xi); f (sk(xi))ijxi 2 A; i = 1; 2; : : : g, where 0 = x1 < x2 < : : : and limi!1 xi = 1. Let s0k : [0; 1] ! [0; 1] be the piecewise linear function connecting neighboured points from fhf (xi); f (sk(xi))ig. Similarly, for each 1 l q, let d0l be the piecewise linear function connecting neighboured points from fhf (xi); f (dl(xi))ig. It can be shown that all s0k and d0l satisfy (20)-(25). Hence, B expanded by all s0k and d0l is a standard Lsp;;dq-chain into which A is embedded. 5 5.1
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 (S1lh) (D1lh)
H' ! ' ' ! R' (S2lh) (D2lh)
V ' ! H' R' ! S' (S3lh) (D3lh)
V 1 :S0 and the following additional deduction rule: (DRlh) from (' ! ) _
infer (h' ! h ) _ ; for each h 2 fV; H; R; Sg: 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 2 X,
H (x) In particular, given the Godel and Lukasiewicz operations respectively de ned 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-de ned operators. 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 modi cation strength.
De nition 5. [
infer (h' ! h ) _ ; for h 2 fs1; :::; sn; d1; :::; dng: The logic Ls2;nd 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 ' = young, it means \slightly young implies not very old ".
Ls2;nd is also a nitary Rasiowa-implicative logic, and its equivalent algebraic semantics is the class of Ls2;nd-algebras.
De nition 6. [
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 ; fc ; c+g; H = fV; H; R; Sg; ), a linguistic logic, denoted Lldh, is an expansion of L with new unary connectives V; H; R; S by the following axioms: (S1ldh) H' ! ' (S2ldh) V ' ! H' (S3ldh) V 1 (D2ldh) R' ! S' (SD1ldh) R' ! :H:' (SD2ldh) S' ! :V :' and the following additional deduction rule: (DRldh) from (' ! ) _
infer (h' ! h ) _ ; for each h 2 fV; H; R; Sg: 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 2 X,
H (x) S (x) x; R (x);
V (x) S (x)
H (x);
V ( x);
V (1) = 1; R (x)
H ( x): Taking into account that 8x 2 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 [ f'g of formulae, `Lldh ' i for every A 2 Lldh and every A-model e of , e(') = 1. (D1ldh) ' ! R'
This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 102.04-2013.21. 7
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 proposed 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 di erent levels of emphasis. For future work, we will study rst-order fuzzy logics with hedges.