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20Multiplying logical values1
GRZEGQRZ MALINQWSKI
abstract. The modern history of many-valuedness starts with Lukasiewicz's construction of three-valued logic. This pioneering, philosophically motivated and matrix based construction, first presented in 1918, was in 1922 extended to n-valued cases, including two infinite ones. Soon several constructions of many-valued logic appeared and the history of the topic became rich and interesting. However, as it is widely known, the problem of interpretation of multiple values is still among vexed questions of contemporary logic.
With the paper, which essentially groups my earlier settlements, from [3], [4], [7] and [8], I intend to put a new thread into discussion on the nature of logical many-valuedness. The topics, touched upon, are: matrices, tautological and non-tautological many-valuedness, Tarski's structural consequence and the Lindenbaum—Wojcicki completeness result, which supports the Suszko's claim on logical two-valuedness of any structural logic. Consequently, two facets of many-valuedness — referential and inferential — are unravelled. The first, fits the standard approach and it results in multiplication of semantic correlates of sentences, and not logical values in a proper sense. The second many-valuedness is a metalogical property of inference and refers to partition of the matrix universe into more than two disjoint subsets, used in the definition of inference.
Keywords: three-valued logic, many-valuedness, matrix, tautology, consequence operation, structurality, logical two-valuedness, Suszko's Thesis, non-fregean logic, logical three-valuedness, inferential many-valued-ness, inferential values.
1 Tautological many-valuedness
A referential construction of a many-valued logic starts with the choice of the sentential language L, i.e. an algebra L = (For, ..., Fm) freely generated by the set of variables Var = {p,q,r,...}. Formulas, i.e. elements of For, are then built from variables using
xThe paper is an extended version of the invited lecture to the international conference "7th Smirnov's Readings in Logic" June 22-24, 2011, Moscow (Russia).
the operations F\,...,Fm representing the sentential connectives. In most cases, either the language of the classical sentential logic
Lk = (For, ^, V, A, o)
with negation (-), implication disjunction (V), conjunction
(A), and equivalence (o), or some of its reducts is considered. Subsequently, one defines an algebra A similar to L and chooses a non-empty subset D of designated elements of the universe of A. Any such interpretation structure
M = (A,D)
is called logical matrix.
Traditionally, the system of sentential logic is defined as the set of all formulas taking for every valuation h (a homomorphism) of L in M
E(M) = {a € For : for every h €Hom(L, A), h(a) € D},
called the content of M. Thus, the classical matrix based on {0,1} and the connectives defined by the classical truth-tables has the form of
M2 = ({0,1}, V, A, o, {1})
Obviously, then E(M2) = TAUT, the set of tautologies. That is why we shall also sometimes refer to E(M) as the set of tautologies of M even if the matrix M is not classical.
If the content of a multiple-element matrix M does not coincide with TAUT, E(M) = TAUT, we say that M defines tautologically many-valued logic. The best known example of this kind is the historically first construction by Lukasiewicz's in [2], here presented as
M3 = ({0,V2,1},V, A, o, {1}) with the connectives defined by the tables:
x — x ^ 0 1/2 1 V 0 /2 1
0 1 0 1 1 1 0 0 /2 1
/2 /2 1/2 /2 1 1 1/2 /2 /2 1
1 0 1 0 1/2 1 1 1 11
A 0 /2 1 0 /2 1
0 0 0 0 0 1 /2 0
/2 0 /2 1/2 1/2 /2 1 /2
1 0 /2 1 1 0 /2 1
Since e.g. p V —p and —(p A —p) take the value /2 when p takes /2, and thus they are not Lukasiewicz tautologies, E(M3) = TAUT.
Subsequently, to see that the separate multiplication of values of the base matrix is not sufficient for getting many-valued logic, consider the following three-element matrix, cf. [7],
N3 = ({0,t, 1}, —, V, A, o, {t, 1}),
x — x ^ 0 t 1 V 0 t 1
0 1 0 1 t 1 0 0 t 1
t 0 t 0 t t t t t 1
1 0 1 0 t 1 1 1 1 1
A 0 t 1 o 0 t 1
0 0 0 0 0 1 0 0
t 0 t t t 0 t t
1 0 t 1 1 0 t 1
It is easy to show, that E(N3) = TAUT. Thus, N3 determines the system of tautologies of the classical logic and it is not tautologically many-valued. To verify that, notice that due to the choice of the set of designated elements {t, 1}, with every ha € Hom(L,A) the classical valuation h* € Hom(L, M2) corresponds in a one-to-one way such that ha € {t, 1} iff h*a = 1.
2 Structurality and its matrix representation
The notion of matrix consequence is a natural generalisation of the classical consequence: a relation \=M C 2For x For is a matrix consequence of M provided that for any X C For, a € For
X \=m a iff for every h € Hom(L, A)(ha € D whenever hX C D).
It is obvious that E(M) = {a : 0 |=M a}. The classical consequence is the matrix consequence of M2, |=M2 . Therefore, if M determines non-classical set of tautologies then |=M = |=M2, and the logic identified with the consequence |=M is not many-valued in the second sense. Again, the Lukasiewicz matrix consequence \=M3 is a good example. On the other hand, the consequence relation determined by the matrix N3 is "entirely" classical, \=n3 = |=m2 .
In the end, let us mention that in the literature one may find examples of matrices, whose content coincides with set of classical tautologies, TAUT, but their consequence relation is non-classical. The matrix S3 is one of the kind:
S3 = ({0, t, 1}, V, A, o, {t, 1}), where
x — x ^ 0 t1 V 0 t 1
0 1 0 1 11 0 0 0 1
t 1 t 1 11 t 0 0 1
1 0 1 0 01 1 1 1 1
A 0 t 1 0 0 t 1
0 0 0 0 0 1 1 0
t 0 0 0 t 1 1 0
1 0 0 1 1 0 0 1
Since t and 0 are indistinguishable by the truth tables in formulas containing the connectives, thus practically in all formulas except the propositional variables, and that both values are distinguished, we obtain E(S3) = TAUT. Accordingly, M2 is the only two-element matrix which might determine \=^3. Simultaneously, \=^3 = |=m2 , since, for example,
{p ^ q,p} |=m2 q while not {p ^ q,p} |=m3 q.
To verify this it simply suffices to turn over a valuation h such that hp = t and hq = 0, cf. [7].
The last example shows that it is indispensable to make a distinction between tautological and consequential many-valuedness. Accordingly, the classical system of tautologies was extended to a
three-valued logic and this extension was assured by rules of inference and not by rejection of logical laws.
The consequence relation has an important algebraic counterpart. With any |=M there may be uniquely associated an operation CnM : 2For — 2For such that
a € CnM (X) if and only if X |=M a.
CnM is called a matrix consequence operation of M. The distinguishing property of the logic generated by M, i.e. |=M and CnM, is its structurality: for any substitution, i.e. an endomorphism of the language, e € END(L),
X |=M a implies eX |=M ea, or equivalently
a € CnM(X) implies ea € CnM(eX).
The concept of structural sentential logic is the ultimate generalisation of the notion of the matrix consequence operation. A structural logic for L is identified with a Tarski's consequence C : 2For — 2For,
(T0) X c C(X)
(ti) C(X) c C(Y) whenever X c Y
(T2) c(c(X)) = C(X),
satisfying the condition of structurality,
(S) eC(X) c C(eX) for every substitution of L.
Structural logics are characterizable through their set of logical laws and schematic rules of inference. The most important property of these logics is their matrix representation: for every such C a class of matrices K exists such that C is the intersection of {CnM : M € K}:
C(X) = n{CnM(X) : M € K}, for any X c For.
The property just mentioned is a version of Lindenbaum-Wojcicki completeness theorem, see [12], which was proved using the Lindenbaum bundle L, i.e. the class of all Lindenbaum matrices of the form (L, C(X)), L = { (L, C(X)) : X c For}. This fact, when combined
with the non-Fregean approach by Suszko [10], importantly subscribes to the problem of our interest.
3 Logical two-valuedness
The generalization of the truth-functionality principle constituting the base of matrix description of the classical logic is at the heart of the matrix method. Accordingly, the search for common features of the both: many-valued and classical, bivalent logics, is natural. The first serious attempts with this respect can be traced back to early 1950-ies. Especially, to Rosser and Turquette project of formalization of a class of finite-valued standard logics, cf. [9]. The main feature of the sentential part of standard logics was their definitional ability to indicate the counterparts of the classical negation, implication, disjunction, conjunction, and equivalence connectives, which with respect to the division between the undistinguished and distinguished sets of values behave as the classical connectives with respect to the falsity and truth. An important continuation of the idea is due to Bloom and Brown, who in [1] introduced and studied classical abstract logics in which the properties of respective connectives are described by the consequence operation of the logic (the idea of such a characterization may be traced back to Tarski).
Accordingly, in the 1970's the investigations of logical formali-zations bore several descriptions of many-valued constructions in terms of zero-one valuations, see e.g. [7]. The approach to the problem by R. Suszko, the author of non-Fregean logic, cf. [10], is the most thoroughly justified. The methodology of this approach will also support our characterization of inferentially many-valued logics.
The base of R. Suszko's philosophy was the distinction between semantic correlates of sentences and their logical values. The author argues that potentially there is no limitation as to the number of correlates of sentences, though there are only two logical values corresponding to Truth and Falsity. In this perspective, all traditionally (i.e. tautologically or consequentially) many-valued logics are also logically two-valued and they may be regarded only as referentially many-valued but not logically many-valued. Recall that the core of the Fregean approach is the so-called Fregean Axiom, see [10], identifying semantic correlates of sentences and their logical values.
Suszko underlines the referential character of homomorphisms associating sentences with their possible semantic correlates (i.e. referents or situations) and sets them against the logical valuations being zero-one-valued functions on For and, thus, setting apart the referential and the logical valuedness. Suszko claims that each sentential logic, i.e. a structural consequence relation, can be determined by a class of logical valuations and thus, it is logically two-valued. The arguments, supporting the Suszko's thesis, are based on the completeness of any structural consequence C with respect to a Lindenbaum bundle.
Given a sentential language L and a matrix M = (A, D) for L, the set of valuations TVM is defined as:
TVm = {th : h € Hom(L, A)}, where
t ( ) = { 1 if h(a) € D
th(a) = \ 0 if h(a) € D.
Consequently, the matrix consequence operation |=M may be described using valuations as follows:
X |=M a iff for every t € TVM t(a) = 1 whenever t(X) c {1}.
The definition of logical valuations may be simply repeated with respect to any structural consequence operation C using its Lindenbaum bundle. Thus, each structural logic (L, C) can be determined by a class of logical valuations of the language L or, in other words, it is logically two-valued.
The justification of the thesis on logical two-valuedness of an important family of logics lacks, however, a uniform description of TVC's i.e. classes of valuations C-adequate, and in each particular case to find TVC is a matter of an elaboration. Below, we discuss the case of finite Lukasiewicz logics.
The original example, given by Suszko, see [11] and [7], presents a relatively easily definable and readable set of logical valuations LV3 for the (—, —) — version of the three-valued Lukasiewicz logic. Accordingly, LV3 is the set of all functions t : For — {0,1} such that for any a,@,j € For the following conditions hold:
(0) t(Y) = 0 or t(-Y) = 0
(1) t(a — P) = 1 whenever t(P) = 1
(2) if t(a) = 1 and t(P) = 0, then t(a — P) = 0
(3) if t(a) = t(P) and t(-a) = t(-P), then t(a — P) = 1
(4) ift(a) = t(P) = 0 and t(-a) = t(-P), then t(a — P) = t(-a)
(5) if t(-a) = 0, then t(--a) = t(a)
(6) if t(a) = 1 and t(P) = 0, then t(-(a — P)) = t(-P)
(7) if t(a) = t(-a) = t(P) and t(-P) = 1, then t(-(a — P)) = 0.
The uniform description of all finite LVn for n ^ 3 requires the use of more compound formulas with the well known one-argument connectives a0,a1,..., an-1 of Rosser and Turquette, definable by means of L-implication and L-negation, cf. [9]. Thus, for any h € Homn
h(ai(a)) =
1 if h(a) = i/n — 1 0 otherwise.
Given n ^ 2, LVn is then the set of all functions t € 2FM such that for any a,P,j € FM the following conditions are satisfied:
(1) t(Y) = t(an-i(Y)); t(-Y) = t(ao(7)).
(2) For each 7 € For exactly one of the following equalities holds: t(ao(Y)) = 1 or t(ai(i)) = 1... or t(an-i(Y)) = 1.
(3) t(ai(Y)) = 1 or t(-ai(i)) = 1; t(ai(-i)) = t(an-i-i(l)).
{0 i = 0 and i = n — 1
t(aj (y)) i = n — 1
t(-aj (y)) i = 0.
(5) t(a — P) = 1 if t(ail (a)) = t(ai2(P)) = 1 and i\ ^ i2, t(ai(a — P)) = 1 if t(ail(a)) = t(ai2 (P)) = 1 where i\ > i2 and i = (n — 1) — ii + i2.
The last description, originally given and thoroughly justified in [4], is algebraic in its spirit, since the function corresponding to Rosser and Turquette connectives, as well as their counterparts are used in several algebraic formalizations of many-valued logic. This naturally suggests that some other classes of logics may have similar two-valued characterization.
4 Inferential three-valuedness
Logical two-valuedness of any structural logic is a direct consequence of the construction of matrices and the division of the sets A of their elements into two complementary subsets of designated and undesignated elements: D and A — D, respectively. And, the 0 — 1 valuations are but characteristic functions of the sets of formulas, which are actually associated, via a homomorphism of the language into the matrix, with designated referents. The conclusion that getting out of logical two-valuedness might be made on the basis of a non-complementary division of the set A of referential values.
In [8] a generalisation of Tarski's concept of consequence operation related on the idea that the rejection and acceptance need not be complementary was proposed. The central notions of the framework are counterparts of the concepts of matrix and consequence relation — both distinguished by the prefix "q" which may be read as "quasi". Where L is a sentential language and A is an algebra similar to L, a q-matrix is a triple
M * = (A,D*,D),
where D* and D are disjoint subsets of the universe A of A (D*nD = 0). D* are then interpreted as sets of rejected and distinguished elements values of M, respectively. For any such M* one defines the relation hM* between sets of formulae and formulae, a matrix q-consequence of M* putting for any X c For, a € For
X |=M* a iff for every h € Hom(L, A)(ha € D whenever hXnD* = 0).
The relation of q-consequence was designed as a formal counterpart of reasoning admitting rules of inference which from non-rejected assumptions lead to accepted conclusions. The q-concepts coincide with usual concepts of matrix and consequence only if D* UD = A, i.e. when the sets D* and D are complementary. Then, the set of rejected elements coincides with the set of non-designated elements.
Sometimes, it is convenient to consider the alternate of q-matrix M* received in result of replacing D* with A — D*. Then, after renaming we consider the matrix M* = (A,D2,Di), where D2 =
A — D* and D1 = D. Note that, then D2 D D1. Accordingly, X \=M* a iff for every h € Hom(L, A)(ha € D1 whenever hX €
This way of expressing the ^-consequence will be used in the next
For every h € Hom(L, A) a function kh : For — {0,1 /2,1}
( 0 if h(a) € D* kh(a) = { /2 if h(a) € A — (D* U D) { 1 if h(a) € D.
may be associated. And, given an inference matrix M* for L we put KVM* = {kh : h € Hom(LA)}, and define \=M* by
X \=M* a iff for every kh € KVM* if kh(X) C {1/2,1}, then kh(a) = 1.
With each relation \=M* one may associate the operation WnM* : 2For — 2For putting
WnM * (X) = {a : X \=m * a}.
This is a kind of a three-valued description of \=M*. Notice that KVM* reduces to TVM and \=M* to \=M when D* U D = A. It is worth emphasising that this is a case only exceptionally: for the
D2).
Section.
D r\D = 0 D*vD * A
D2 □ A
ordinary (structural) consequence relation. In general, the description remains three-valued. Therefore a q-logic (L,WnM* ) is either two- or three-valued.
■0 if d (j d= a
The following example proves that the three-valued q-logics exist, cf. [8]. Take the three-element q-matrix
L3 = ({0, % 1}, V, A, O, {0}, {1}),
where the connectives are defined as in the Lukasiewicz three-valued logic. Then, for any p € Var, it is not true that {p} hM* p. To see this, it suffices to consider the valuation sending p into /2.
The more striking is perhaps the fact that even logics generated by some two-element q-matrices may be logically three-valued.
Though different, the two approaches coincide on contents. Note, that
For any q-matrix M* = (A, D*, D) and a corresponding matrix
M = (A,D), wum* (0) = cum(0) = E(M).
This is a very important observation. It means that any logical system may equally well be extended to logically two-valued logic (L,Cum) or to a three-valued logic (L,Wum*). Obviously, depending on the quality and cardinality of M the two kinds of extensions may define different logics. Moreover, in several cases it is also possible to define two (or more) different inferential extensions of a given system. The idea was applied in [5] to make a distinction between two "indistinguishable" modal connectives of the four-valued modal system of Lukasiewicz.
The g-framework is general. In [8] an inference operation of which WnM* is a prototype was introduced and studied. An operation W : 2For — 2For is a q-consequence operation provided that for every X,Y C For
(W1) W(X U W(X)) = W(X)
(W2) W(X) C W(Y) whenever X C Y.
W is called structural if for any substitution e € End(L)
(S) eW(X) C W(eX).
Where M* is any inference matrix, WnM* is structural. In turn, all Lindenbaum's tools may be adopted to structural inference operations W to exactly the same effect. Thus, the bundle of Lindenbaum's inference matrices
Wx = ( For , For — (X U W(X)) , W(X))
may be used to prove, cf.[8], that
For every structural inference operation W there is a class K of inferential matrices such that
Wnx(X) = f]{WnM* (X) : M* € K}.
Following the line of constructing the valuations corresponding to the classes of matrices, here applied to q-matrices, we may get a necessary description of any structural q-logic (L,W). Accordingly, we conclude that any such construction is logically two- or three-valued.
5 More inferential values?
Extending the inferential approach to more values and thus getting logical n-valuedness for n > 3 seems easy: apply the method of construction described in Section 4 and divide matrix universe into more than three mutually disjoint subsets. However, the next step, i.e. construction of an appropriate matrix g(n)-consequence is not evident. On the other hand, no natural lattice-theoretic combination of q-consequences is possible, since g-consequence operations on a given language form a lattice, see [6].
In [3] a partial solution of the problem is provided. Using special bundle of finite linear matrices with the unary functions "labelling" appropriate subsets, the construction of a compositional operation was defined, which in the "inferential" terms is more than 3 valued.
The construction is based on a finite algebra En = (En, fi, f2, ■ ■ ■, fm) where u > 2 and En = {1,2,3, ■ ■ ■, u}. Next, it is required that among the functions of En, primitive or definable, there are unary functions 51,52, ■ ■ ■, 5n:2
ô ()_ / n if x > i + 1 dl(x) = \ 1 if x<i + 1.
Let L = (En,Fi,F2,... ,Fm) be the language to which En is similar. Assume that Dk = {k,k — 1,...,n} and Di = {l, l + 1,..., n}, for k < l and, finally take the ^-matrix:3
MOii = (En,Dk ,Di),
Consider its ^-inference,
X h M°la iff for every h e Hom(L,En) (If h(X) ç Dk then h(a) e Di),
and the matrix Mn = (En, {n}) and its consequence =M
X =M a iff for every h e Hom(L,En) (If h(X) ç {n}, then h(a) e {n}).
Let ôk (X) =df {5k(fi) : /3 e X} and notice that
X h°Mki a iff ôk(X)=M 5i(a),
and, finally, that
QnMki(X) = {a : X h°Mk i a} = {a : ôk(X) =M ôi(a)}.
The last equation shows two descriptions of sets QnMk l (X) and thus, descriptions of the matrix ^-consequence operation QnMk l.
2Such functions are definable in standard logics, cf. [9], a family of which contains most of the known finite-valued calculi. They are also widely used in modern formulations of Post algebras.
3In the second formulation provided in Section 4.
The crucial concept of a multidimensional matrix and its inference is defined using a decreasing sequence of non-empty subsets of En, Ds-i,...,D1:
Ds-i D Ds-2 D ... D Di, such that En — Ds-1 = 0.
The structure
Ms = (En, Ds-1 ,Ds-2,...,Di)
is referred to as an s dimensional q-matrix. The notion of an s dimensional q-matrix is a direct generalization of the concept of q-matrix; every q-matrix is three dimensional and "standard" matrices are two dimensional. Next to that, as compatible with the appropriate consequence-like operation Qs defined below, the partitions correspond directly to logical values, see [3] for details.
The notion of s dimensional q-matrix inference which would keep the correspondence preserved to a higher degree is then defined using the bundle Ms of q-matrices associated with Ms :
Ms = {(En, Ds-1, Ds-2), (En, Ds-2, Ds-3), ■■■, (En, D2, Di)}, or, equivalently, by
Ms = {Ms-2,Ms-3,...,Mi },
where Ms-i =df (En, Ds-i+i, Ds-i), i = l,...,s — 2.
Now, if Qi = WnMs-i is the matrix q-consequence of Ms-i, then the operation Qs : 2For — 2For defined as superposition of Qs-2, Qs-3, ...,Q2,Qi, Qs = Qs-2 ◦ Qs-3 ◦ ... ◦ Q2 ◦ Qi, which for every set of formulas X
Qs(X) = Qs-2(Qs-3(. . . (Q2(Qi(X))) . . .)),
satisfies the starting requirements.
The definition of multiple-element valuations is slightly involved. To outline the idea of it, fully discussed in [3], we shall consider the case of 5-dimensional matrix q-consequence W determined by a matrix M5 = (En, D4, D3, D2, Di).
As one may expect, the logic determined by W should be in-ferentially five-valued. One may also expect that, as in the three-valued case of q-consequence, the valuations should be in correspondence with homomorphisms from L into En, i.e. the referential assignments sending formulas into the matrix universe. Accordingly, valuations are functions mapping appropriate subsets of En into the five-valued set of logical (inferential) values, say into {0,1/4,2/4,3/4,1}. The subsets of En, mapped in subsequent values are respectively: En — D4,D4 — D3,D3 — D2,D2 — D1,D1. The following diagram
depicts the association. 6 Final remarks
The modern conceptual treatment of many-valuedness was possible due to matrices introduced to logic on the turn of 20th century. The matured matrix methodology, as a part of a general theory of logical calculi, cf. [12], essentially extended the possibilities.
In the paper we discussed two kinds of matrix based referential many-valuedness: tautological and consequential. The first case occurs, when the multiplication of elements of matrix universe results in a smaller set of logical truths, i.e. set properly included in the set of all classical tautologies, see Section 1. Otherwise, it is possible to get many-valuedness using the matrix consequence: conservatively, saving the classical set of tautologies, or non-conservatively. The first case requires the use of "non-classical rules" of inference, see Section 2.
The concept of many-valuedness becomes more involved on the grounds of the theory of structural consequence operations, since logics thus defined are often characterizable with the classes of matrices, cf. [12]. So, sometimes it may be difficult to say to what degree a given logic is (referentially, or algebraically) "many-valued". The situation essentially changes in view of the Suszko's methodology inspired by the non-Fregean approach in which denotations of sentences i.e. referential, or algebraic values are different entities from logical values. Accordingly, due to the Suszko's Thesis, every structural logic is logically two-valued.
The construction of q-consequence was the first step towards logical many-valuedness, or more precisely, to inferential three-valu-edness. The methodological background for structural q-logics has already been elaborated: the q-consequence is described through modification of Tarski's conditions for consequence and the q-matrix methodology as a generalisation of the standard matrix approach, is provided. Finally, the Lindenbaum-Wojcicki's - like completeness result leads to the conclusion that any structural q-logic is logically two- or three-valued. Accordingly, the logical three-valuedness departs naturally from the division of the matrix universe into three subsets and the (ST) counterpart says that any inference based on a structural q-consequence may have a bivalent or a three-valued description, cf. Section 3.
The discussion of possibilities for further exploration of the idea of logical n-valuedness for n > 3 showed that the only evident step to be made consists of a division of the matrix universe into more than three subsets. Therefore, instead of the general approach to logically many-valued inference, only finite linear matrices with one-argument functions "labelling" subsets of elements of the matrix were considered. Unfortunately, even for the case, which is important, getting uniform description of properties of logically more-valued consequence-like relation is not easy. So, one may only say, that the construction presented in Section 4 proves that logically n-valued logics do exist. The problem of getting general description and elaboration of methodology for such devices still remains open.
It should be also underlined, that getting a general axiomatization of structural s-dimensional q-consequence, as well as its completeness proof, seems very hard indeed. Perhaps the algebraic hints given
in Section 3 may help in making further steps toward even more general theory of many-valued inference.
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