A New Algebraic Version of Monteiro’s Four-Valued Propositional Calculus ()

Aldo Victorio Figallo, Estela Bianco, Alicia Ziliani

Departamento de Matemática, Universidad Nacional del Sur, Bahía Blanca, Argentina.

Instituto de Ciencias Básicas, Universidad Nacional de San Juan, San Juan, Argentina.

**DOI: **10.4236/ojpp.2014.43036
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Departamento de Matemática, Universidad Nacional del Sur, Bahía Blanca, Argentina.

Instituto de Ciencias Básicas, Universidad Nacional de San Juan, San Juan, Argentina.

In the XII Latin
American Symposium on Mathematical Logic we presented a work introducing a
Hilbert-style propositional calculus called four-valued Monteiro propositional
calculus. This calculus, denoted by M_{4}, is introduced in terms of the binary
connectives (implication), → (weak
implication), ∧ (conjunction) and the unary ones (negation) and ▽ (modal
operator). In this paper, it is proved that M_{4} belongs to the class of standard systems of
implicative extensional propositional calculi as defined by Rasiowa (1974). Furthermore, we
show that the definitions of four-valued modal algebra and M_{4} -algebra are equivalent and, in addition,
obtain the completeness theorem for M_{4}. We also introduce the notion of modal
distributive lattices with implication and show that these algebras are more
convenient than four-valued modal algebras for the study of four-valued
Monteiro propositional calculus from an algebraic point of view. This follows
from the fact that the implication → is one of its basic
binary operations.

Keywords

Mathematical Logic, Hilbert-Style Propositional Calculus, Non-Classical Logics, Four-Valued Monteiro Propositional Calculus

Share and Cite:

Figallo, A. , Bianco, E. and Ziliani, A. (2014) A New Algebraic Version of Monteiro’s Four-Valued Propositional Calculus. *Open Journal of Philosophy*, **4**, 319-331. doi: 10.4236/ojpp.2014.43036.

1. Preliminaries

In 1978,

A four-valued modal algebra is an algebra of type such that the reduct is a distributive lattice with greatest element 1 satisfying these conditions:

(L1),

(L2),

(L3),

(L4).

From the definition, it follows that A is a De Morgan algebra ( Monteiro, 1960; Birkhoff, 1967 ). Besides, Loureiro (1982, 1983a) proved que that the variety of these algebras is generated by the algebra, where

1) is the De Morgan algebra such that and, being not comparable;

2) The operation is defined by and for all.

In what follows, we will denote by the category of four-valued modal algebras and their corresponding homomorphisms.

For the notions of universal algebra including distributive lattices, De Morgan algebras and category theory used in this paper we refer the reader to ( Kalman, 1958 ; Birkhoff, 1967 ; MacLane, 1971 ; Burris & Sankappana- var, 1981 ).

The structure of the paper is as follows: In Section 1, we develop a Hilbert-style propositional calculus which we call four-valued Monteiro propositional calculus and denote by. Besides, we prove that belongs to the class of standard systems of implicative extensional propositional calculi ( Rasiowa, 1974 ). Furthermore, we show that the notions of four-valued modal algebra and -algebra are equivalent. Finally, we demonstrate that is consistent and that the completeness theorem holds. From the equivalence established above, we conclude that from any algebra we have that is an -algebra where and and conversely, from any -algebra we obtain a four-valued modal algebra by defining. These statements and the fact that can be defined from allow us to assert that and the variety generated by are polynomially equivalent. In Section 2, we introduce the variety of modal distributive lattices with implication and we show that the category of these algebras and their corresponding homomorphisms is equivalent to. Thus, the category theory notably simplify the proof that should be performed to obtain this result by direct calculations. Hence, we determine a new equational descrip- tion of four-valued modal algebras which is more convenient than the latter to study from an algebraic point of view, since the implication is one of its basic binary operations.

2. Four-Valued Monteiro Propositional Calculus

The main aim of this section is to describe the propositional calculus and show that it has four-valued modal algebras as the algebraic counterpart. The terminology and symbols used here coincide in general with those used in ( Rasiowa, 1974 ).

Let be a formalized language of zero order such that in the alphabet the set

1) of propositional variables which they will be denoted by, is enumerable;

2) is empty;

3) L_{1} contains two elements denoted by and called negation sign and modal operator sign, respectively;

4) contains three elements denoted by, and called conjunction sign, weak implication sign and implication sign, respectively;

5) contains two elements denoted by.

Let F be the set of all formulas over A^{0}. For any in, we shall write for short instead of.

We assume that the set of logical axioms consists of all formulas of the following form, where are any formulas in:

(A1),

(A2),

(A3),

(A4),

(A5),

(A6),

(A7),

(A8),

(A9),

(A10),

(A11),

(A12),

(A13),

(A14),

(A15),

(A16),

(A17),

(A18),

(A19),

(A20),

(A21),

(A22),

(A23),

(A24),

(A25),

(A26),

(A27),

(A28),

(A29),

(A30).

The consequence operation in is determined by the set and by the following rule of inference:

(R1), (

The system thus obtained, will be called four-valued Monteiro propositional calculus. It is worth mentioning that the above connectives are not independent, however, we consider them for simplicity. We shall denote by the set of all formulas derivable in. As usual, if we shall say that is a theorem of and we shall write or.

In Lemma 1.1 we summarize the most important rules and theorems necessary for further development.

Lemma 1.1. In the following rules and theorems hold:

(R2),

(R3),

(T1),

(T2),

(R4),

(R5),

(R6),

(T3),

(T4),

(R7),

(R8),

(R9),

(R10),

(R11),

(R12),

(T5),

(R13),

(R14),

(T6),

(T7),

(T8),

(T9),

(T10),

(T11),

(R15),

(T12),

(T13),

(R16),

(T13),

(R17),

(R18),

(R19).

Proof. We only prove (R15), (R16) and (R17).

(R15):

1),

2), [(1), (R11), (A3), (r1)]

3), [(1), (R11), (A4), (r1)]

4), [(A7), (R4)]

5), [(3), (4), (r1)]

6), [(T11), (R4)]

7), [(3), (6), (r1)]

8), [(A9), (7), (R7)]

9). [(5), (8), (R9), (R12)]

(R16):

1),

2),

3), [(1), (R11), (A4)]

4), [(2), (R11), (A3)]

5), [(2), (R11), (A4)]

6), [(5), (A16), (r1)]

7), [(6), (T13), (r1)]

8), [(A18), (3), (4), (r1)]

9). [(7), (8), (R9), (R12)]

(R17):

1),

2),

3), [(1), (R11), (A3)]

4), [(2), (R11), (A3)]

5), [(2), (R11), (A4)]

6), [(T4), (3), (r1)]

7), [(T14), (3), (r1)]

8), [(T9)]

9), [(8), (T1), (r1)]

10), [(7), (A12), (R7)]

11), [(10), (A11), (R7)]

12), [(11), (9), (r1)]

13), [(6), (12), (R9), (R12)]

14), [(T2), (4), (r1)]

15), [(5), (R4)]

16), [(A11), (15), (R7)]

17), [(A12)]

18), [(16), (17), (R7)]

19), [(18), (A13), (R7)]

20), [(14), (19), (R9), (R12)]

21). [(13), (20), (R13)] □

Theorem 1.1. The propositional calculus belongs to the class of standard systems of implicative extensional propositional calculi.

Proof. We have to prove that conditions (s1) to (s8) in ( Rasiowa, 1974 ) are satisfied. Clearly, (s1) and (s2) are verified. Besides, (s3), (s4), (s5) and (s6) follow from (T5), (R10), (R13) and (R14), respectively. On the other hand, taking into account (R15) and (R16), we have that (s7) holds. Finally, (R17), (R18) and (R19) allow us to conclude (s8). □

Next, our attention is focused on stating the relationship between four-valued modal algebras and - algebras ( Rasiowa, 1974 ). Lemma 1.2 will be fundamental for this purpose.

Lemma 1.2. In the following theorems hold:

(T15),

(T16),

(T17),

(T18),

(T19),

(T20).

(T21),

(T22),

(T23),

(T24),

(T25),

(T26),

(T27).

Proof. We only prove (T17), (T20), (T22) and (T27).

(T17):

1), [(A4)]

2), [(1), (R8)]

3), [(2), (R4)]

4), [(T16)]

5), [(4), (R4)]

6), [(3), (5), (r1)]

7), [(A2)]

8), [(7), (R6)]

9), [(A3), (8), (r1)]

10), [(9), (R4)]

11). [(6), (10), (r1)]

(T20):

1), [(A19)]

2), [(1), (R4)]

3), [(2), (T8), (r1)]

4), [(T7)]

5). [(3), (4), (R7)]

(T22):

1), [(T18)]

3), [(A3)]

4), [(T17)]

5), [(A10)]

6), [(4), (R2)]

7), [(3), (5), (r1)]

8), [(2), (6), (R9)]

9). [(7), (1), (r1)]

(T27):

1), [(T26)]

2), [(1), (R8)]

3), [(T7)]

4), [(3), (R4)]

5), [(4), (T25), (r1)]

6). [(5), (2), (r1)] □

Remark 1.1. It is worth noting that if is a formula derivable in, then for every valua- tion of in every -algebra.

Proposition 1.1. Let. Then is an -algebra, where and are defined as follows:

,

.

Proof. We shall prove that conditions (a1) to (a4) in ( Rasiowa, 1974 ) hold. Indeed, taking into account the definitions of and we have that (a1) and (a2) are satisfied. On the other hand, let be such that. Then, we get (1) and (2). From (1), (A2) and (2), (A14) we have that and respectively. These assertions and the fact that if and only if and allow us to conclude that and so, (a3) holds. Besides, if, then by an analogous argument to that used in the proof of (a3), we get, from which we infer (a4). □

Proposition 1.2. Let be an -algebra. Then where .

Proof. To show that is a De Morgan algebra, by ( Marona, 1964 ) it suffices to prove that conditions (M1) and (M2) are satisfied. By virtue of Remark 1.1 and (a4) in ( Rasiowa, 1974 ), (M2) follows from (A23) and (M1) from (A3), (A19), (T19), (T20), (R9) and (R12). Moreover, following an analogous reasoning and taking into account (A24) we infer (L3); besides, (T22), (T23), (T24), (T27), (R9) and (R12) allow us to conclude (L4). Hence, the proof is complete. □

From Propositions 1.1 and 1.2 we infer:

Theorem 1.2. The notions of -algebra and four-valued modal algebra are equivalent.

Let be the binary relation on defined as follows:

if and only if and in.

Then, is a congruence relation on and determines an equivalence class which will be denoted by 1. Moreover, is an -algebra ( Rasiowa, 1974 ) and therefore from Proposition 1.2, we conclude:

Theorem 1.3..

On the other hand, since is consistent, from ( Rasiowa, 1974 ) and Theorem 1.2 we have that the completeness theorem for holds, which is included in.

Theorem 1.4. Let be a formula of. Then the following conditions are equivalent:

1) is derivable in;

2) is valid in every -algebra;

3), where is the canonical valuation in the algebra.

3. Categorical Equivalence between and

In this section we introduce the notion of modal distributive lattice with implication and we prove some pro- perties of these algebras which allow us to show the announced categorical equivalence.

Definition 2.1. A modal distributive lattice with implication (or -algebra) is an algebra of type where is a bounded disitributive lattice satisfying the following identities:

(I1),

(I2),

(I3),

(I4),

(I5),

(I6),

(I7),

(I8),

(I9),

(I10),

(I11),

(I12).

In Proposition 2.1 we show some properties of -algebras which will be useful in what follows.

Proposition 2.1. Let be an -algebra. Then the following identities hold:

(I13),

(I14),

(I15),

(I16),

(I17),

(I18),

(I19),

(I20),

(I21),

(I22),

(I23),

(I24) implies,

(I25) implies,

(I26) if and only if, ,

(I27) implies,

(I28),

(I29).

Proof. We only prove (I22) and (I26).

(I22): From (I21), (I5), (I3) and (I12) we have that

.

(I26): Let. Then, from (I4), (I1), and (I14) we conclude that. Besides, (I1) allows us to infer. Conversely, from the hypothesis, (I13) and (I12) we have that

, and so,. □

Let be an -algebra. Then, we define, where for every. Furthermore, given the -algebras, and a homo- morphism, we define by.

Lemma 2.1 and Propositions 2.2, 2.3 and 2.4 are fundamental in order to prove Theorem 2.1.

Lemma 2.1. Let be an -algebra. Then the following properties are verified:

(I30),

(I31),

(I32),

(I33),

(I34),

(I35),

(I36) implies,

(I37) implies,

(I38),

(I39) implies.

Proof. We only prove (I30) and (I39).

(I30): From (I5), (I17), (I20) and (I4) we have that and. Besides, (I23), (I6) and (I1) allow us to infer that . In addition, from (I8), (I23), (I6) and (I26) we have. By these assertions and (I26) we conclude the proof.

(I39): From (I4), (I5) and (I31) we get that

.

Then, by (I21), (I36), (I14) and (I2) we conclude that. On the other hand, from (I35), (I38), (I19) and (I36) we have that . Hence, the above assertions and (I26) allow us to conclude that. □

Proposition 2.2. is a four-valued modal algebra.

Proof. We only have to prove that satisfies conditions (L1)-(L4).

(L1): Taking into account (I34), (I33), (I8) and (I15) we infer that

.

then, by (I19), (I21), (I3) and (I30) we have that

.

(L2): It is a direct consequence of (I39) and (L1).

(L3): From (I23) and (I10) we conclude that. This identity and (I10) allow us to infer that.

(L4): It follows from (I6) and (I30). □

Proposition 2.3. is a full and faithful functor between -algebras and four-valued modal algebras.

Proof. It follows from Proposition 2.2 and the definition of. □

Proposition 2.4. Every four-valued modal algebra is isomorphic (equal) to for some -algebra.

Proof. Let, where for all. As properties (I1)-(I12) hold in, then is an -algebra. In order to complete the proof, we must show that. If we de- fine, by Proposition 2.2 it follows that is a four-valued modal algebra. In addition, for every, , since this identity holds in. □

Theorem 2.1. is a categorical equivalence between and.

Proof. It is a direct consequence of Propositions 2.3, 2.4 and Theorem 1 of ( MacLane, 1971 ). □

Theorem 2.1 and the results on Section 1 confirm that -algebras are the algebraic counterpart of the four-valued Monteiro propositional calculus.

Conflicts of Interest

The authors declare no conflicts of interest.

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