Grzegorz Malinowski

REFERENTIAL AND INFERENTIAL MANY-VALUEDNESS

The development of the method of logical matrices at the turn of XIX Century made it possible to define the concept of many-valued logic. However, the problem of interpretation of logical values in addition to truth and falsity is still among the most controversial question of contemporary logic. The aim of the talk is to present two faces of the problem of many-valuedness: referential and inferential. In the first approach manyvaluedness may be received as the result of multiplication of semantic correlates of sentences, and not logical values. In the former many-valuedness (more precisely, three-valuedness) is the metalogical property of inference which lead from non-rejected assumptions to accepted conclusions.

1. Logical matrices and tautological manyvaluedness

A generic construction of a many-valued logic starts with the choice of the sentential language L which may be shown as an algebra L = (F or, F_{1}, …, F_{m}) freely generated by the set of sentential variables V ar = {p, q, r, …}. Formulas, i.e. elements of F or, are then built from variables using the operations F_{1}, …, F_{m} representing the sentential connectives. In most cases, either the language of the classical sentential logic

L_{k} = (F or, ¬, →, ∨, ∧, ↔)

with negation (¬), implication (→), disjunction (∨), conjunction (∧), and equivalence (↔), or some of its reducts is considered. Subsequently, one defines a multiple-valued algebra A similar to L and chooses a non-empty subset of the universe of A, D ⊆ A, of designated elements. The interpretation structures

M = (A, D)

are called logical matrices.

Given a matrix M for a language L, the system E(M) of sentential logic is defined as the content of M i.e. the set of all formulas taking for every valuation h (a homomorphism) of L in M. Thus

E(M) = {α ∈ F or : for every h ∈Hom(L, A), h(α) ∈ D}.

In the case when M is the classical matrix based on {0, 1}, i.e.

M_{2} = ({0, 1}, ¬, →, ∨, ∧, ↔, {1})

with the connectives are defined by the classical truth-tables, E(M) is the set of tautologies TAUT. That is why in the sequel we shall sometimes refer to E(M) as the set of tautologies of a matrix M even if it is not classical.

When it is the case that the content of a multiple-element matrix M does not coincide with TAUT, E(M) ≠ TAUT, we say that the matrix in question defines tautologically many-valued logic. The examples of such logics are well known from the literature. Every scholar in logic knows the historically first construction of the three-valued L logic by Łukasiewicz, whose matrix is

M_{3} = ({0,^{1} /_{2}, 1}, ¬, →, ∨, ∧, ↔, {1})

and the connectives are defined by the following tables:

\begin{matrix}x & ¬ x \\ 0 & 1 \\ ^{1}/_{2} & ^{1}/_{2} \\ 1 & 0 \end{matrix}  \begin{matrix}→ & 0 & ^{1}/_{2} & 1 \\ 0 & 1 & 1 & 1 \\ ^{1}/_{2} & ^{1}/_{2} & 1 & 1 \\ 1 & 0 & ^{1}/_{2} & 1 \end{matrix}  \begin{matrix}∨ & 0 & ^{1}/_{2} & 1 \\ 0 & 0 & ^{1}/_{2} & 1 \\ ^{1}/_{2} & ^{1}/_{2} & ^{1}/_{2} & 1 \\ 1 & 1 & 1 & 1 \end{matrix}

\begin{matrix}∧ & 0 & ^{1}/_{2} & 1 \\ 0 & 0 & 0 & 0 \\ ^{1}/_{2} & 0 & ^{1}/_{2} & ^{1}/_{2} \\ 1 & 0 & ^{1}/_{2} & 1 \end{matrix}  \begin{matrix}↔ & 0 & ^{1}/_{2} & 1 \\ 0 & 1 & ^{1}/_{2} & 0 \\ ^{1}/_{2} & ^{1}/_{2} & 1 & ^{1}/_{2} \\ 1 & 0 & ^{1}/_{2} & 1 \end{matrix}

Obviously, E(M3) ≠ TAUT, since e.g. p ∨ ¬p and ¬(p ∧ ¬p) do not belong to the set of tautologies of Łukasiewicz logic.

1.1. Consider the matrix

M_{3} = ({0, t, 1}, ¬, →, ∨, ∧, ↔, {t, 1}) for L_{k} with the operations defined by the following tables:

\begin{matrix}x & ¬ x \\ 0 & 1 \\ t & 0 \\ 1 & 0 \end{matrix}  \begin{matrix}→ & 0 & t & 1 \\ 0 & 1 & t & 1 \\ t & 0 & t & t \\ 1 & 0 & t & 1 \end{matrix}  \begin{matrix}∨ & 0 & t & 1 \\ 0 & 0 & t & 1 \\ t & t & t & 1 \\ 1 & 1 & 1 & 1 \end{matrix}  \begin{matrix}∧ & 0 & t & 1 \\ 0 & 0 & 0 & 0 \\ t & 0 & t & t \\ 1 & 0 & t & 1 \end{matrix}

\begin{matrix}↔ & 0 & t & 1 \\ 0 & 1 & 0 & 0 \\ t & 0 & t & t \\ 1 & 0 & t & 1 \end{matrix}

We claim that this three-valued matrix determine the system of tautologies of the classical logic. To verify that it suffices to notice that due to the choice of the set of designated elements {t, 1}, with each hα ∈ Hom(L, A) the classical valuation h^{∗} ∈ Hom(L, M_{2}) corresponds in a one-to-one way such that hα ∈ {t, 1} iff h^{∗}α = 1. Thus, the logic under consideration in neither sense is many-valued.

2. Matrix consequence and structural logics

The notion of matrix consequence being a natural generalisation of the classical consequence is defined as follows: relation ⊧ M ⊆ 2^{F or} × F or is a matrix consequence of M provided that for any X ⊆ F or, α ∈ F or

X ⊧_{M} αif and only if for everyh ∈ Hom(L, A)(hα ∈ DwheneverhX ⊆ D).

Notice, that E(M) = {α : ∅ ⊧_{M} α}. Therefore, if a matrix determines non-classical set of tautologies then obviously ⊧_{M} does not coincide with the consequence relation of the classical logic, i.e. with ⊧_{M_{2}}. As before, we may consider the three-element matrix of Łukasiewicz to have an example. In what follows, we note also that the consequence relation determined by the matrix in 1.1 is classical.

The next example of three-element matrix logic is surprising and shows that there are matrices determining as its content the same set of classical tautologies, but its consequence relation is non-classical.

2.1. Consider the matrix

M_{3} = ({0, t, 1}, ¬, →, ∨, ∧, ↔, {t, 1})

for L_{k} with the operations defined by the following tables:

\begin{matrix}x & ¬ x \\ 0 & 1 \\ t & 1 \\ 1 & 0 \end{matrix}  \begin{matrix}→ & 0 & t & 1 \\ 0 & 1 & 1 & 1 \\ t & 1 & 1 & 1 \\ 1 & 0 & 0 & 1 \end{matrix}  \begin{matrix}∨ & 0 & t & 1 \\ 0 & 0 & 0 & 1 \\ t & 0 & 0 & 1 \\ 1 & 1 & 1 & 1 \end{matrix}  \begin{matrix}∧ & 0 & t & 1 \\ 0 & 0 & 0 & 0 \\ t & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 \end{matrix}

\begin{matrix}↔ & 0 & t & 1 \\ 0 & 1 & 1 & 0 \\ t & 1 & 1 & 0 \\ 1 & 0 & 0 & 1 \end{matrix}

Taking into account that t and 0 are indistinguishable by the truth tables in formulas containing the connectives, thus practically all formulas except the propositional variables, and that the both values are distinguished, we obtain E(M_{3}) = TAUT. Accordingly. M_{2} is the only two-element matrix which might determine ⊧_{M_{3}}. Simultaneously, ⊧_{M_{3}}≠⊧_{M_{2}}, 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

The example shows that it is reasonable to make a distinction between tautological and consequential many-valuedness. The classical system (of tautologies) was extended to a three-valued logic and the very property was assured by rules of inference rather than by its logical laws.

With every ⊧_{M} there may be uniquely associated an operation Cn_{M} : 2^{F or} → 2^{F or} such that

α ∈ Cn_{M}(X) if and only if X ⊧_{M} α.

called a matrix consequence operation of M.

The concept of structural sentential logic is the ultimate generalisation of the notion of the matrix consequence operation. A structural logic for a given language L is identified with a Tarski’s consequence C : 2^{F or} → 2^{F or},

(T0) X ⊆ C(X)
(T1) C(X) ⊆ C(Y) whenever X ⊆ Y
(T2) C(C(X)) = C(X),

satisfying the condition of structurality,

(S) eC(X) ⊆ C(eX) for every substitution of L.

Structural logics are characterizable through their set of logical laws and schematic (sequential) rules of inference. The most important property of these logics is their matrix characterisation: for every such C a class of matrices \underline{K} exists such that C is the intersection of \{Cn_{M} : M ∈ \underline{K}\} i.e. for any X ⊆ F or

C(X) = ∩\{Cn_{M}(X) : M ∈ \underline{K}\}.

The problem of their many-valuedness is thus reducible to the problem of many-valuedness of matrix consequence relations (or operations).

3. Logical two-valuedness of structural logics

In the 1970’s the investigations of logical formalizations bore several descriptions of many-valued constructions in terms of zero-one valuations. The interpretations associated with these descriptions shed new light on the problem of logical many-valuedness. One of the best justified and general approaches is due to R. Suszko, the author of non-Fregean logic, cf. [9].

The base of R. Suszko’s philosophy of logic was the distinction between semantic correlates of sentences and their logical values. In this perspective the traditionally many- valued logics (tautologically or consequentially) may be regarded as only referentially many-valued and not necessarily logically many-valued. Recall that one of the central points of the Fregean approach in [1], the so-called Fregean Axiom, identified semantic correlates of sentences with their logical values.

Suszko [9] draws attention to the referential character 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 F or and, thus, differentiates 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 argument supporting the Suszko thesis is the completeness of any structural censequence C with respect to a Lindenbaum bundle, which is the class of all Lindenbaum matrices of the form (L,C(X)) with X ⊆ F or.

Given a sentential language L and a matrix M = (A, D) for L, the set of valuations TV_{M} is defined as:

TV_{M} = {t_{h} : h ∈ Hom(L, A)},

where

t_{h}(x) = \{ \begin{matrix}1 & if & h(α) ∈ D \\ 0 & if & h(α) ∉ D. \end{matrix}

Consequently, the matrix consequence operation ⊧_{M} may be described using valuations as follows:

X ⊧_{M} α if and only if for every t ∈ T V_{M}t(α) = 1 whenever t(X) ⊆ {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 that states logical two-valuedness of an important family of logics lacks a uniform description of T V_{C}’s i.e. the respective classes of valuations. The task is in each particular case a matter of an elaboration. An example of a relatively easily definable and readable set of logical valuations is LV_{3}, the class adequate for the (¬, →) – version of the three-valued Łukasiewicz logic, cf. [2]. LV_{3} is the set of all functions t : F or → {0, 1} such that for any α, β, γ ∈ F or the following conditions hold:

(0) t(γ) = 0 or t(¬γ) = 0

(1) t(α → β) = 1 whenever t(β) = 1

(2) if t(α) = 1 and t(β) = 0, then t(α → β) = 0

(3) if t(α) = t(β) and t(¬α) = t(¬β), then t(α → β) = 1

(4) if t(α) = t(β) = 0 and t(¬α) ≠ t(¬β), then t(α → β) = t(¬α)

(5) if t(¬α) = 0, then t(¬¬α) = t(α)

(6) if t(α) = 1 and t(β) = 0, then t(¬(α → β)) = t(¬β)

(7) if t(α) = t(¬α) = t(β) and t(¬β) = 1, then t(¬(α → β)) = 0,

cf. [9].

4. Inferential many-valuedness

In [7] a generalisation of Tarski’s concept of consequence operation related upon 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^{∗} ∩ D = ∅). D^{∗} are then interpreted as sets of rejected and distinguished elements values of M, respectively. For any such M^{∗} one defines the relation ⊢_{M∗} between sets of formulae and formulae, a matrix q-consequence of M^{∗} putting for any X ⊆ F or, α ∈ F or

X ⊢_{M^{∗}} α if and only if for every h ∈ Hom(L, A)(hα ∈ D whenever hX ∩ D^{∗} = ∅).

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^{∗} ∪ D = 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.

For every h ∈ Hom(L, A) let us define a three-valued function k_{h} : F or → {0,^{1} /_{2}, 1} putting

k_{h}(α) = \{ \begin{matrix}0 & if & h(α) ∈ D^{∗} \\ ^{1}/_{2} & if & h(α) ∈ A − (D^{∗} ∪ D) \\ 1 & if & h(α) ∈ D. \end{matrix}

Given a q-matrix M^{∗} for L let KV_{M} = {k_{h} : h ∈ Hom(L, M^{∗}); we get the following three-valued description of the q-consequence relation ⊢_{M^{∗}}:

X ⊢_{M^{∗}} α if and only if for every k_{h} ∈ KV_{M}(k_{h}(X) ∩ {0} = ∅ implies k_{h}(α) = 1).

It is worth emphasising that this description in general is not reducible to the two-valued description possible for the ordinary (structural) consequence relation. As the later property may be interpreted as logical twovaluedness of logics identified with the consequence, we may say that a q-logic is logically either two or three valued. Moreover, the three-valued q-logics exist, cf. [7]. The example below shows that it is the case.

3.1. Consider the three-element q-matrix

£_{q}3 = ({0,^{1} /_{2}, 1}, ∼, ⇒, ∨, ∧, ≡, {0}, {1}),

where the connectives are defined as in the Łukasiewicz three-valued logic. Then, for any p ∈ V ar, it is not true that {p} ⊢_{M^{∗}} p. To see this, it suffice to consider the valuation sending p into ^{1}/_{2}.

The more striking is perhaps the fact that even logics generated by some two-element q-matrices may be three-valued. This is illustrated by our last example:

3.2. Let us consider the two-element algebra

A_{2} = ({0, 1}, ¬, →, ∨, ∧, ↔),

with the operations defined by the classical truth-tables of negation, implication, disjunction and equivalence. Next, let us consider the following two q-matrices:

M_{1} = (A_{2}, ∅, {1}),

M_{0} = (A_{2}, ∅, {0}).

The q-consequence relations of M_{1} and M_{0} are such that for any X ⊆ F or, α ∈ F or

X ⊢_{M_{1}} α if and only if for every h ∈ Hom(L, A_{2})hα = 1,

X ⊢_{M_{0}} α if and only if for every h ∈ Hom(L, A_{2})hα = 0.

Thus, in the first case a formula α is a q-consequence of any set of formulas, whenever it is a tautology. In the second case α is a contradictory formula.

The standard description of ⊢_{M_{1}} in terms of {0,^{1} /_{2}, 1}-valuations k_{h} is then defined in such a way that for every α ∈ F or, k_{h}(α) = 1 iff α ∈ T AUT, k_{h}(α) = ^{1}/_{2} otherwise; for no formula k_{h} takes the value 0.

Similarly, X ⊢_{M0} α whenever k_{h}(α) = 0, where k_{h}(α) = 0 iff α is contradictory and k_{h}(α) = ^{1}/_{2} otherwise.

5. Final remarks

On the ground of the general theory of interpretation of sentential languages and structural logics defined on these languages, cf. [10], a discussion of the concept of many-valuedness was given. Following the ways traced by R, Suszko I distinguished between two its types: referential and inferential.

The inferential approach was based on the notion of q-matrix and the matrix q- consequence. First observe that

4.1. For any q-matrix M^{∗} = (A, D^{∗}, D) and a corresponding matrix M = (A, D), W n_{M^{∗}}(∅) = Cn_{M}(∅) = E(M).

This means that any logical system may equally well be extended to logically two-valued logic (L, Cn_{M}) or to a three-valued logic (L, W n_{M^{∗}}). Obviously, depending on the quality and cardinality of M both kinds of extensions may take different shapes, thus defining different logics. Moreover, in several cases it is also possible to define two (or more) different inferential extensions of a given system thus receiving different q-logics. The idea was applied in [4] to get the framework permitting to make a distinction between two “indistinguishable” modal connectives of the four-valued modal system of Łukasiewicz.

The logical two-valuedness in the sense of the paper is related to the division of the universe of a matrix into two subsets: designated and undesignated. The logical three-valuedness set forth in Section 3 is clearly mirrored in the construction of q-matrices: the universe of any q-matrix is divided into three subsets. In both cases logical valuations are characteristic functions of the divisions.

4.2. For some inferentially three-valued logics based on three-valued algebras, referential assignments i.e. algebraic homomrphisms h ∈ Hom(L, A) and logical valuations k_{h} ∈ KV_{M} do coincide (compare 3.1). Such logics are in a sense extensional with respect to three logical values and in a sense satisfy a generalised version of the Fregean postulate, which identifies semantic correlates with logical values, cf. [5]

4.3. The q-framework is general. Similarly as in the case of matrix consequence it can be related to the theory of the so-called q-consequence operation W : 2^{F or} → 2^{F or} satisfying the following postulates

(T1) W (X) ⊆ W (Y) whenever X ⊆ Y

(T2) W (X ∪ W (X)) = W (X),

and, possibly, the condition of structurality,

(S) eW (X) ⊆ W (eX) for every substitution of L.

An extensive study of the framework is given in [7], cf. also [5]. The result corresponding to the Suszkos thesis is that each \underline{q-logic} i.e. structural q-consequence operation \underline{is logically two or three-valued}.

References

[1] Frege, G., Über Sinn und Bedeutung. Zeitschrift für Philosophie und philosophische Kritik C, 1892, 25-50.

[2] Lukasiewicz, J., O logice trójwartoociowej, Ruch Filozoficzny, 5, 1920, 170-171. English tr. On three-valued logic [in:] Borkowski, L. (ed.) Selected works, North- Holland, Amsterdam, 87-88.

[4] Malinowski, G., Inferential extensions of Lukasiewicz modal logic, an invited lecture to the Conference Lukasiewicz in Dublin, University College Dublin, Department of Philosophy, Dublin 7 - 10 July 1996.

[5] Malinowski, G., Inferential many-valuedness [in:] Wolenski, J. (ed.) Philosophical logic in Poland, Synthese Library, 228, Kluwer Academic Publishers, Dordrecht, 1994, 75-84.

[6] Malinowski, G., Many-valued logics, Oxford Logic Guides, 25, Clarendon Press, Oxford, 1993.

[7] Malinowski, G., Q-consequence operation, Reports on Mathematical Logic, 24, 1990, 49-59.

[8] Suszko, R., Abolition of the Fregean Axiom [in:] Parikh, R. (ed.) Logic Colloquium, Symposium on Logic held at Boston, 1972-73. Lecture Notes in Mathematics, vol. 453, 1972, 169-239.

[9] Suszko, R., The Fregean Axiom and Polish Mathematical Logic in the 1920s,StudiaLogica, XXXVI (4), 1977, 377-380.

[10] Wójcicki R., Theory of logical calculi. Basic theory of consequence operations, Synthese Library, 199. Kluwer Academic Publishers, Dordrecht, 1988.

Department of Logic University of Łódź Poland gregmal@krysia.uni.lodz.pl