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On axiomatizations of general many-valued propositional calculi

Arto Salomaa Turku Centre for Computer Science Joukahaisenkatu 3–5 B, 20520 Turku, Finland asalomaa@utu.fi

Abstract We present a general setup for many-valued propositional , and compare -table and axiomatic stipulations within this setup. Results are obtained concerning cases, where a finitary axiomatization is (resp. is not) possible. Related problems and examples are discussed.

1 Introduction

Many-valued systems of are constructed by introducing one or more truth- values between truth and falsity. Truth-functions associated with the logical connectives then operate with more than two truth-values. For instance, the truth-function D associated with the 3-valued disjunction might be defined for the truth-values T,I,F (true, intermediate, ) by

D(x, y) = max(x, y), where T > I > F.

If the truth-function N associated with is defined by

N(T ) = F,N(F ) = T,N(I) = I, the law of the excluded middle is not valid, provided “” means that the truth-value t results for all assignments of values for the variables. On the other hand, many-valuedness is not so obvious if the axiomatic method is used. As such, an ordinary axiomatization is not many-valued. Truth-tables are essential for the latter. However, in many cases it is possi- ble to axiomatize many-valued logics and, conversely, find many-valued models for systems. In this paper, we will investigate (for propositional log- ics) interconnections between such “truth-value stipulations” and “axiomatic stipulations”. A brief outline of the contents of this paper follows. In the next two sections we define formally the basics of the two stipulations and present some simple results of the related notion of a consequence. Section 4 applies the notions

1 in a classical framework, and also studies the case where the truth-functions constitute a functionally complete . Section 5 (resp. Section 6) investigates cases where a finite model is (resp. is not) possible for an axiom system. We also pay attention to the minimal cardinality of the model. The last two sections contain some preliminary discussion and open problems. Remark. The topic of this paper was chosen because I had many discussions with Sandu Mateescu about many-valued logic, often in connection with mem- ories of Gr.C. Moisil. Moisil was one of the early investigators in the field, still often quoted in the literature, [6, 11, 2, 4].

2 Many-valued propositional logics. Truth-table stipulation

Our approach in defining many-valued propositional logics (calculi) is very gen- eral and suitable for all systems considered in the past. In many respects we follow the definitions of , [13]. We first define the notion of a propositional logic.

Definition 1 Consider two sets of primitive signs: • A denumerably infinite set {p, q, r, . . .} called (propositional) variables,

• { a1 ak } ≤ ≤ A finite set C1 ,...,Ck called connectives. For 1 i k, we say that ai Ci is an ai-place connective. A finite sequence of primitive signs is a well-formed formula (wff)if it either ai consists of a , or else equals Ci (α1, . . . , αai ), for some i, 1 ≤ i ≤ k, where each αj, 1 ≤ j ≤ ai, is a wff. The set of all well-formed formulas is denoted by WFF . Let VA and NVA be two of WFF , both closed under alphabetical variance of variables. The triple (W F F, V A, NV A) is called a propositional partial logic if VA and NVA are disjoint. If in addition WFF = VA ∪ NV A, then the triple is called a propositional logic (or a .)

Some additional remarks are in order. Well-formed formulas contain also imprimitive signs (parentheses, comma) not made explicit in the definition. We could also add logical constants (such as T for truth) but this is not needed for our purposes. The notation VA (resp. NVA) is introduced with the word valid (resp, non-valid) in mind. For n ≥ 2, the numbers 1, . . . , n are referred as truth-values. For i ≥ 1, any function mapping the Cartesian power {1, . . . , n}i into the set {1, . . . , n} is called an i-place truth-function in n values. Truth-functions are associated with connectives in a natural way. For instance, the function of two variables defined by f(1, 1) = f(1, 2) = f(2, 1) = 1, f(2, 2) = 2

2 is associated with the (2-valued) disjunction. (The semantic order of the truth- values will be defined below.) We are now in the position to define the notion of a many-valued propositional logic.

Definition 2 A propositional logic (W F F, V A, NV A) is an n-valued proposi- tional logic, n ≥ 2, if the following four conditions are satisfied.

1. With each propositional variable p is associated a distinct numerical vari- able x.

ai ≤ ≤ 2. With each connective Ci , 1 i k, is associated an ai-place truth- ai function in n values, called the associate of Ci . 3. The following condition is satisfied for all wff’s α, β, γ. Whenever β is the result of uniformly substituting a variable p in α with γ, then the associate of β is the result of uniformly substituting the associate of p with the associate of γ in the associate of α.

4. There is a natural number d, 1 ≤ d < n, such that VA equals the set of wff’s whose associates assume a truth-value less than or equal to d, for all assignments of values for the associates of their variables.

A propositional logic is a many-valued propositional logic if it is an n-valued propositional logic, for some n. The truth-values 1, . . . , d (resp. d + 1, . . . , n) are referred to as designated (resp. undesignated) truth-values.

As an example, we consider some connectives in the Lukasiewiczthree-valued logic, [1]. (It was already used as an example in the Introduction.) The truth- values are 1, 2, 3, and 1 is the only designated truth-value. The truth-functions associated to the 1-place connective F1 and 2-place connective F2 are defined by, respectively,

f1(x) = 3 − x + 1, f2(x, y) = min(x, y).

The connectives F1 and F2 correspond to the 2-valued negation and disjunction, respectively. The two truth-functions yield, by Definition 2, a 3-valued proposi- tional logic (W F F, V A, NV A). Without going into the study of the set VA in this case, we mention that the wff F2(p, F1(p)), corresponding to the 2-valued law of the excluded middle is not in VA. This follows because

f2(2, f1(2)) = f2(2, 2) = 2.

Semantic studies concerning the truth-values lie outside the scope of this paper.Lukasiewicz interprets the intermediate truth-value 2 as ”possible”, ”undecided” or ”unknown”. There are also various possibilities to generalize the ordinary 2-valued connectives to the many-valued case. The reference [9] contains a fairly exhaustive classification.

3 In Definition 2 we have introduced a truth-value stipulation for the set VA. This stipulation emphasizes the fact that we are dealing with a many-valued logic. Another common procedure for characterizing the set VA is the axiomatic method. Certain wff’s, , are chosen as assertable. Moreover, certain operations on wff’s, rules of inference, are specified. In this axiomatic stipulation the set VA is defined to be the closure of the set of axioms under the rules of inference. In the axiomatic stipulation many-valuedness is not as clear as in the truth- vale stipulation. The following two general problems are basic in this set-up. 1. Is a given many-valued propositional logic axiomatizable? In other words, can we choose a finite AX of WFF , as well as a finite set OP of closure operations on WFF , such that the set VA equals the closure PR, “provable” wff’s, of AX under the operations in OP ? 2. Does there exist a finite many-valued model for a given axiom system? More specifically, given an axiom system (W F F, AX, OP ), we are looking for integers n and d, and truth-tables for the connectives in WFF such that defining VA as the closure of AX under the operations in OP yields a many-valued propositional logic. The purpose of this paper is to present some results concerning these two problems.

3 Axiomatic stipulation

We now return to the formal definition of the notions discussed above, as well as some other related notions. The set WFF of well-formed formulas is defined as in Definition 1.

Definition 3 A triple AS = (W F F, AX, OP ) is referred to as an axiom sys- tem if AX is a finite subset of the set WFF , and OP is a finite set of closure operations on the set WFF such that alphabetic variance is one of the opera- tions in OP . The closure PR of AX under the operations in OP , in symbols CL(AX, OP ), is called the set of formulas provable under the axiom system AS. The axiom system AS is called inconsistent if PR = WFF . Otherwise, AS is consistent. If AS is consistent but CL(AX ∪{α}, AS) = WFF , for every wff α not in PR), then AS is complete. Consider two wff’s α and β, as well as a subset U of WFF . The set of consequences of U in the axiom system AS is defined by

CONS(U, AS) = CL(AX ∪ U, OP ).

If CONS(U, AS) = WFF , we say that U is inconsistent with AS. The wff’s α and β are contradictory in AS if

CONS({α}∪{β}, AS) = WFF and CONS({α}, AS)∩CONS({β}, AS) = PR.

4 The notions were introduced without any reference to particular connectives, such as negation and implication. The definition of the latter is problematic in many-valued logic, [9]. The following obvious lemma describes some basic properties of the operation CONS in this general set-up. The is denoted by ∅. Lemma 1 Consider an axiom system AS = (W F F, AX, OP ) and subsets U and V of the set WFF . Then the following relations hold. 1. U ⊆ CONS(U, AS).

2. PR ⊆ CONS(U, AS). 3. If U ⊆ AX then CONS(U, AS) = PR. In particular, CONS(∅, AS) = PR. 4. CONS(CONS(U, AS), AS) = CONS(U, AS).

5. If U ⊆ V then CONS(U, AS) ⊆ CONS(V, AS). 6. CONS(U ∩ V, AS) ⊆ CONS(U, AS) ∩ CONS(V, AS).

7. CONS(U ∪ V, AS) = CONS(U ∪ CONS(V, AS), AS). 8. CONS(U ∪ V, AS) = CONS(CONS(U, AS) ∪ CONS(V, AS), AS).

9. CONS(U, AS) ∪ CONS(V, AS) ⊆ CONS(U ∪ V, AS). 10. If uniform substitution is the only operation in OP then

CONS(U, AS) ∪ CONS(V, AS) = CONS(U ∪ V, AS).

The following definition describes the (eventual) interconnection between the truth-table and axiomatic stipulations.

Definition 4 A many-valued propositional logic MVL = (W F F, V A, NV A) is a model for an axiom system AS = (W F F, AX, OP ) if VA = PR. If such a model exists, AS is said to possess a finite model. Conversely, AS is said to be an axiomatization of MVL. The logic MVL is axiomatizable if an axiomati- zation AS exists for it.

We have assumed that the set AX if finite. This assumption is essential because, without it, every MVL is axiomatizable.

5 4 Classical axiomatizations. A general construc- tion

We consider first the 2-valued propositional logic L2 = (W F F, V A, NV A), where d = 1 and WFF is determined by the 2-place connective ⇒ and the 1-place connective ∼ whose truth-functions are those of the material implica- tion and negation, respectively. In what follows we often omit unnecessary parentheses. We also often write pCq instead of C(p, q), for a 2-place connective C.

We now define an axiom system AS2 = (W F F, AX, OP ) as follows. The set WFF is the same as in L2. The set AX consists of the three wff’s 1. p ⇒ (q ⇒ p) (Affirmation of the consequent.) 2. (r ⇒ (p ⇒ q)) ⇒ ((r ⇒ p) ⇒ (r ⇒ q)) (Self-distributive law of implica- tion.) 3. (∼ p ⇒∼ q) ⇒ (q ⇒ p) (Converse law of .)

(We have indicated in parentheses the meaning of the three wff’s in .) The set OP consists of the following two rules of inference.

1. Uniform substitution for the variables. 2. : From the wff’s α ⇒ β and α to infer the wff β.

The axiom system AS2 is investigated in great detail in [3]. The results can be summarized, in our terminology, as follows.

Theorem 1 The logic L2 is axiomatizable. The logic L2 is a model for the axiom system AS2. Hence, the axiom system AS2 possesses a finite model.

Consider next the 2-valued propositional logic LE based on the sole con- nective equivalence, ≡, and the associated 2-valued truth-function. (See [3] for more details.) Consider, further, the axiom system ASE, where the set of wff’s equals that of LE, the set of operations consists consists of the uniform substi- tution for the variables and modus ponens with respect to ≡, and there are the following two axioms. 1. (p ≡ q) ≡ (q ≡ p),

2. (p ≡ (q ≡ r)) ≡ ((p ≡ q) ≡ r),

It can be shown that ASE is an axiomatization of LE and, consequently, ASE possesses a finite model. Moreover, a wff α is provable (and belongs to the set VA of LE) exactly in case every variable occurs in α an even number of times. The reader might want to show the provability of the wff

p ≡ (p ≡ (q ≡ (q ≡ (r ≡ r)))).

6 We will now prove a general result concerning the problem of finding an axiomatization for a many-valued propositional logic. For this purpose we need the notion of . Truth-functions some Cartesian power of the set N = {1, 2, . . . , n} into N. Clearly, any composition of truth-functions is itself a truth-function. (We omit here the formal definition of a composition. The notion is understood in a very general sense. For instance, the ternary function

f(z, g(f(x, g(y, z)), g(y, f(x, z)))) is a composition of the two binary functions f(x, y) and g(x, y).)

Definition 5 A set F of n-valued truth-functions is functionally complete if every n-valued truth-function, independently of the number of variables, equals a composition of some functions in F. If F consists of a single function f, then f is termed a Sheffer function.

We need the following result due to J. S lupecki, [12, 1]. Lemma 2 Let n ≥ 2 and d, 1 ≤ d ≤ n − 1 be arbitrary. If the n-valued truth- functions associated with the connectives of a many-valued propositional logic Ln,d (with d designated values) form a functionally complete set, then the logic Ln,d possesses a complete axiomatization.

Theorem 2 Given any n and d such that n ≥ 2 and 1 ≤ d ≤ n − 1, there is an n-valued propositional logic Ln,d, with d designated values and a single connective C(p, q), possessing a complete axiomatization, where modus ponens with respect to C is one of the rules of inference

Proof. Let C be the binary connective whose truth-function c(x, y) is defined by c(n, y) = y + 1, c(x, y) = n, for all y and all x ≤ n − 1. (Addition is carried out modulo n.) The function c(x, y) is a Sheffer function. The proof of this fact is omitted, since criteria of functional completeness, [8, 10], lie outside the scope of this paper. By Lemma 2, we still have to prove that modus ponens with respect to C can be taken as one of the rules of inference. But αCβ assumes never a designated value if α assumes a designated value. Hence, adding modus ponens with respect to C to the rules of inference does not produce new provable wff’s. 2 We still apply Theorem 2 to 2-valued propositional logic. The connective C is now “joint rejection”: C(p, q) is true exactly in case both p and q are false. Negation and material implication can be expressed in terms of C as follows:

∼ p = C(p, p), p ⇒ q = C(C(C(p, p), q),C(C(p, p), q)).

From the axiom system AS2 of Theorem 1 we now obtain a complete axiomati- zation of the propositional logic L2,1. The additional rule of modus ponens with

7 respect to C does not alter the set PR, because of the premises α and C(α, β) are never both in PR.

Axiomatizations have been obtained in a number of cases, where the set of truth-functions is not necessarily complete. J.B. Rosser and A. Turquette, [7], have constructed an axiomatization for many-valued propositional logics in case certain particular connectives are present or definable in terms of the given ones. For instance, consider a 2-place connective C and k 1-place connectives Di, 1 ≤ i ≤ k, as well as the corresponding truth-functions c and di, 1 ≤ i ≤ k. Suppose c(x.y) assumes an undesignated value exactly in case x is designated and y undesignated, and each di(x), 1 ≤ i ≤ k, assumes a designated value exactly in case x = i. Then an axiomatization can be constructed. M. Wajsberg, [14], has shown that a many-valued propositional logic with a single connective C(p, q) is axiomatizable, provided that the associated truth- function c(x, y) satisfies the following condition. Each of the functions

c(c(x, y), c(c(y, z), c(x, z))), c(c(y, z), c(c(x, y), c(x, z))), c(c(y, y), c(x, x)) assumes a designated value, for all values of x, y, z.

5 Axiom systems possessing a finite model

We now prove that a consistent axiom system AS, where uniform substitution for the variables is one of the rules of inference, possesses a finite model if the number of wff’s not provable under AS is finite except for alphabetical variance. (When we say that a set U of wff’s is finite except for alphabetical variance, we mean that every infinite subset U1 of U contains wff’s which are alphabetical variants of some other wff’s in U1.) Theorem 3 A consistent axiom system AS = (W F F, AX, OP ), where uniform substitution for the variables is one of the operations in OP , possesses a finite model if the set WFF − PR is finite except for alphabetical variance.

ai ≤ ≤ Proof. Let Ci , 1 i k be all the connectives appearing in WFF . ai (As before Ci is an ai-place connective.) We will determine numbers n and ≤ ≤ − ai ≤ ≤ d, 1 d n 1, as well as n-valued truth-functions ci , 1 i k, such that the n-valued propositional logic Ln,d is a model of AS. Choose a set B = {α1, α2, . . . , αt} of wff’s such that B contains exactly one from each of alphabetical variants in WFF − PR. By the hypothesis, t is finite. Since a sole variable is not in PR, we may assume that α1 = p, and every αi, i > 1, begins with a connective. Since infinitely many variables are available, we may assume also that no variable appears in two αi’s, and no variable appears twice in any αi. The number n of truth-values will be specified below, and d = n−1. We now define the truth-functions associated with the connectives in such a way that

8 each of the wff’s αi, 2 ≤ i ≤ t, assumes the value n, for at least one assignment of values for the variables, whereas a designated truth-value results in all other cases. In order to avoid in the definitions, we number all well- formed proper parts of the wff’s α with the numbers 2, . . . , m, and define the function values accordingly. We then choose n = m + 2. (We need an additional designated truth-value 1 to make the definitions complete.) We illustrate the method with an example. Assume that one of the α’s is the wff C(C(p1,B(p2)),B(C(p3, p4))). (Recall that all variables were chosen to be different.) The well-formed proper parts of this wff are

p1, p2, p3, p4,B(p2),C(p1,B(p2)),

C(p3, p4),B(C(p3, p4)).

Number the parts, in thus order, with the numbers 2,..., 9. This gives rise to the following values of the truth-functions c(x, y) and b(x).

b(3) = 6, b(8) = 9, c(2, 6) = 7, c(4, 5) = 8, c((7, 9) = n.

This procedure is carried out for every α. All values of the truth-functions not specified in this fashion are defined to be the value 1. (This includes the values for the truth-functions associated with connectives not appearing in any in the wff’s α.) The construction guarantees that Ln,n−1 is a model for the axiom system AS. 2

The construction in Theorem 3 usually gives an unnecessarily large value for n. We now present an example, where the value of n obtained by the construction cannot be reduced. Consider the axiom system AS = (W F F, AX, OP ), where uniform substi- tution for the variables is the only operation in OP , and

AX = {B(p),C(B(p), q),C(p, B(q)),C(C(p, q), r),C(p, C(q, r)),C(p, p).}

(Observe that alphabetic variance is a special case of uniform substitution.) In the notation of Theorem 3 we now have t = 2 and α2 = C(p, q). This is seen as follows. All wff’s having nested occurrences of the connective C are in PR, by the fourth and the fifth axiom (and uniform substitution). Consequently, by the first three axioms, all wff’s containing the connective B are in PR. Since C(p, p) is an axiom, the only wff’s in WFF − PR are (apart from alphabetical variance) p and C(p, q). The well-formed proper parts of C(p, q) are p and q. Following the proof of Theorem 3, We obtain n = 4 and d = 3. The truth-function b(x) is the constant 1, whereas c(2, 3) = 4, c(x, y) = 1, otherwise.

9 We could also choose d = 1 or d = 2 in this example. Assume that a 3-valued model exists for AS. A case analysis shows that a always arises. Assume first that 1 is the only designated truth-value. Then b(x) = 1, for all x. Consequently,

c(b(x), y) = c(1, y) = c(x, b(y)) = c(x, 1) = 1, for all x and y. We have also c(2, 2) = c(3, 3) = 1. Therefore, either c(2, 3) or c(3, 2) is undesignated. By symmetry we may assume that c(2, 3) is undesig- nated. If c(2, 3) = 2, then

c(c(2, 3), 3) = c(2, 3) = 1, by the axiom C((Cp, q), r). If c(2, 3) = 3, then

c(2, c(2, 3)) = c(2, 3) = 1, by the axiom C(p, C(q, r)). A contradiction arises in both cases. Assume, secondly, that both 1 and 2 are designated truth-values. If b(2) = 1, the values c(1, y) and c(x, 1) are designated for all x and y. Hence c(2, 3) = 3 or c(3, 2) = 3, and we may continue as above. If b(2) = 2, we conclude that c(2, y) and c(x, 2) are designated for all x and y. This implies that c(1, 3) = 3 or c(3, 1) = 3. Again we assume by the symmetry that the former alternative holds. A contradiction also now arises because, by axiom C(p, C(q, r), the value c(1, c(1, 3)) = c(1, 3) is designated.

6 Cases where a finite model is not possible

We will prove in this section two results concerning cases where no finite model is possible for an axiom system. The results are extensions of those by Turquette, [7], and G¨odel,[5]. The example at the end of the preceding section showed that a finite model can exist, although C(p, p) is one of the axioms and uniform substitution one of the rules. However, if nothing else is available, then no finite model exists. Theorem 4 Consider an axiom system AS = (W F F, AX, OP ), where

AX = {C1(p, p),C2(p, p),...,Ck(p, p)}, k ≥ 1, and uniform substitution is the only operation in OP . (Apart from Ci, 1 ≤ i ≤ k, the set WFF may contain other connectives.) Such an AS possesses no finite model.

Proof. Define recursively a sequence of wff’s αi, i = 1, 2,..., as follows:

α1 = C1(p, p), αi+1 = C1(αi, αi), i ≥ 1.

10 Assume that an n-valued model exists for AS, and consider the truth-function c1(x, y) associated with the connective C1(p, q). Denote by v(x, i) the truth- value assigned by c1 to αi when the truth-value x is assigned to the variable p. Thus, v(x, 2) = c1(c1(x, x), c1(x, x)), for all x. For a fixed i, v(x, i) is a function mapping the set {1, . . . , n} into itself. Since the total number of such functions is nn, there are numbers r and s, r < s, such that v(x, r) = v(x, s), for all x. Clearly, C1(αs, αs) is in PR. This implies that C1(αr, αs) (and also C1(αs, αr)) is in PR, which is not the case. The contradiction concludes the proof. 2

Ideas similar to the ones used in our next theorem have been applied in the proof of the non-existence of a finite model in case of some well-known propositional logics, such as the intuitionistic propositional logic. We need two technical notions for the statement of the theorem.

Definition 6 A 2-place connective G in an axiom system is termed disjunction- like if, for all wff’s α, β where α ∈ PR, both G(α, β) ∈ PR and G(β, α) ∈ PR. For a disjunction-like G and an arbitrary 2-place connective C, the i-complete (G, C)-wff γi(G, C) is defined by double as follows. i ≥ γ2(G, C) = C(p1, p2), γi+1(G, C) = G(δi+1, γi(G, C)), i 2, where 1 ≥ j+1 j δk = C(p1, pk), k 2, δk = G(C(pj+1, pk), δk).

It is not difficult to see that γi(G, C) is the “G-disjuction” of all wff’s C(pr, ps), 1 ≤ r < s ≤ i. For instance,

γ3(G, C) = G(G(C(p2, p3),C(p1, p3)),C(p1, p2)).

Theorem 5 Consider a disjunction-like connective G and an arbitrary 2-place connective C in an axiom system AS. Define the wff’s γi(G, C), i ≥ 2, as above. Assume, further, that C(p, p) ∈ PR. If γr(G, C), r ≥ 2, is not in PR, and an n-valued propositional logic is a model for AS, then n ≥ r. Consequently, AS possesses no finite model if infinitely many wff’s γi(G, C) lie outside PR.

Proof. Assume the contrary: γr(G, C) is not in PR, and an n-valued propo- sitional logic, n < r, is a model for AS. Consider the truth functions associated with the connectives G and C, and an arbitrary assignment of truth-values for the r variables in γr(G, C). Since n < r, two of the variables get the same value and, hence, the corresponding “disjunctive clause” gets a designated truth-value. (Recall that C(p, p) ∈ PR.) By the properties of G it follows that γr(G, C) gets a designated truth-value. Since the assignment was arbitrary, we conclude that γr(G, C) ∈ PR, a contradiction. 2

11 7 Truth-functions making a wff assertable

When dealing with the two main problems discussed in this paper, one is often confronted with the following task. Given a wff α, find all functions that make α assertable. In other words, if the functions in question are chosen to be truth- functions of the connectives in α, then α will get a designated value, for any assignment of truth-values for the variables. (We assume that n and d are fixed.) In some cases the solution is easy, for instance if α = C(p, p). A converse task is the following. Given a function f(x, y), find all wff’s α, built up using a single binary connective, such that f makes α assertable. We hope to return to these problems in another paper. We mention here only a simple lemma dealing with the case that α involves occurrences of only one connective, a 2-place connective C(p, q). A trivial solution is to check through 2 all the nn 2-place functions. A simple method can be obtained if the given function f(x, y) is doubly increasing in the following sense. For any x, y, z where x > y, we have f(z, x) ≥ f(z, y) and f(x, z) ≥ f(y, z). The proof of the following lemma is omitted.

Lemma 3 Assume that f(x, y) is doubly increasing and g(x, y) satisfies the inequality g(x, y) ≤ f(x, y), for any x and y. Then whenever f(x, y) makes a wff α assertable, also g(x, y) makes α assertable.

8 Conclusion

We have studied some interconnections between the truth-table and axiomatic stipulations. It is likely that many of the results can be strengthened. For in- stance, in Lemma 2 and its consequences full functional completeness is not nec- essary, as was already pointed out, and Theorem 3 can eventually be extended to other assumptions concerning the set WFF − PR. It would be interesting to find a hierarchy of axiom systems, based on the minimal cardinalities of their models.

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