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Title On the Study of Many-valued logics (1)

Author(s) Miyama, Toru

Editor(s)

Citation 大阪府立工業高等専門学校研究紀要, 1979, 13, p.47-57

Issue Date 1979-11-30

URL http://hdl.handle.net/10466/13127

Rights

On the Study of Many-valued logics I

T6ru MI yA MA *

(Received September 3, 1979)

Abstract

For the purpose of grasping at the present condition of many-valued logics, we researchinto the results of many-valued logics obtained in the recent years. Consequently, we look for-

ward to get the guiding principles of the future study of many-valued logics. In this paper, we

research into the results obtained chiefly in the 1960's.

1. Introduction

Recently, the literatures of mathematics are growing rapidly; Mathematical Reviews inthe 1960's inserted about 140,OOO papers in mathematics. Under the circumstances, it becomes

too hard for mathematcians to cover not only the whole ofmathematics but their fields. They

are apt to victimize the current of their fields or other fields if they cover only the current

topics in their field. And they are apt to deal with the papers within the views of their groups,

which is according to their local and personal scales of the originality. Consequently, somemathematicians innocently investigate and present the old results as the original ones without

knowing that they had already been presented.

It is not always effective to study mathematics without the sufficient bibliographicalinvestigations. It interferes with ,the true growth of mathematics and the birth of the original

papers. The prominent papers, which mark a new epoch, may appear from the circumstancesfermented by many ordinary papers. We should find the interesting subjects by knowing thesituations of the investigations. And we should concentrate oureffortsupon the original and

usefu1 subjects by avoiding duplicate efforts. We should make every effort in order to escape 'from such circumstances traced to the rapid growth of the literatures of mathematics. It isimportant to get a timely and sufficient bibliography.

Mathematical Reviews (MR) and Zentralblatt fur Mathematik (Zb) etc. cover the worldresearch literatures in mathematics by means of critical reviews. We can make use of thesejournals. But we need various informations of investigations. So we should make various bibli-

ographies on fair terms with the views ofvarious demands.

With the view mentioned above, we attempt to research the field of many-valued logics in

the 1960's by utilizing MR. Through this attempt, we have sarne problems that MR and Zbhave;one is to clear out the backlog ofunchecked works, and another is an evaluation whichputs the works into a perspective with regard to related works. Specially, it is difficult to

evaluate the works without the subjective judgement. We make our efforts to avoid theseproblems as well as possible. In the future, we attempt to improve ones gradually. As thesupplementary problem, we are also concerned about diacritical marks printed in references.

* DepartmentofGeneralEducation.

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And we suggest the following systems in response to the rapid growth of the literaturesthrough this attempt. The editorial boards ofjournals should demand to authors of treatisesthat they insert some key words and the accurate subject classification code (e.g. AMS,MOS)

on their treatises' footnotes; Fundamenta Informaticae (Annales Societatis MathematicaePolonae) and Mathematica Japonicae etc. have already done. In the future,MR and Zb amongothers should supply the individual information services for individual demands by utilizing

computers and various information systems.

2. Many-valued logics in the 1960's

2.1 . Summary .

179 treatises referred chiefly in the 1960's are classfied to the following currents; prop-

erties of connectives, finite many-valued logics,infinite many-valued logics and the rest. General-

ly, investigations in the 1960's have the tendency to handle and extend the concepts andpropertieS which hold in otherlogics,specially in 2-valued logics. These methods are seman-tical rather than syntactical, and there are some papers where (boolean) algebraic and topo-logical methods are utilized. Most of papers investigate structual concepts and properties in

many-valued logics, and there are few papers whose subjects are applications ofmany-valuedlogics.

A. Rose presented many papers in the 1960's. Specially we are interested in the following treatises;(20),(21),(30),(151), (159).

(20) is a brief survey in set theory with [O, 1] -valued logic as a basis. A main problem is

the consistency of the axiom schema of comprehensionin this logic,which is related to Russell's

paradox. (21) is a semantical survey in continuous logic. Continuous logic contains finite many-valued logics.And topological concepts are utilized with respect to truth values andconnectives.

(30) is a historical note of many-valued logic. The authors offer hitherto unpublishedevidence that in 1909 Pejrce extended the matrix method to triadic logic. This upsets the

accepted opinion that Post (1920) and Lukasiewicz (l921) attributed to extend the matrixmethod to 3-valued logic.

(151) contains the informations on the question whether the set-theoretic antinomies can

be eliminated by modifying the underlying propositional calculus. The author shows that asuitable modification of the Russell's paradox remains in the n-valued Lukasiewicz calculus.

On the other hand, one obtains a consistent system by using the infinite-valued Lukasiewicz

calculus and by restricting the comprehension axiom to quantifier-free formulas.

(159) generally represents finite many-valued logics ofmatrix-type. The representations of

various types are transferable to this system. And this system does not always depend a desig-

nated value. This system is extended to continuous logic by the topological methods.

2 .2 . koperties of connectives.

The treatises in this current are classfied to the following subjects;functional completeness,

precompleteness, Sheffer function, representation of connectives, self dual connective (m-alset), logical computer, variable functor and the rest.

Functional c6mpleteness is discussed in the following papers; (8), (22)<24), (28), (33),(97), (127), (138), (145), (148), (176), (178). Most of these give the necessary and sufficient

conditions to be functionally complete. Some of these treat propositional constants.

Precompleteness is discussed in (35)<37) and (61). This is the concept generalized fromfunctional completeness; a class A is precomplete if it is not complete but ifA U{f} ,where f

is any connective which is not in A , is complete. (36) is done in countabiy-valued logic.

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Sheffer function is discussed in the following papers; (1), (27), (37), (40), (59),(126),

(133), (147), (175). Most of these give the examples to be Sheffer function. Some of thesediscuss the number of these functions. (126) and (!33) are done in infinite-valued logics.

Representation of connectives is discussed in the following papers;(2),(10),(14),(25),

(38),(45),(54),(65),(128),(156),(174). Self dual connective (m-al set) is discussed in the following papers; (108)<110), (112),

(113),(115),(116). Logical computer is discussed in (31), (32) and (114).

Variable functor is discussed in (117)<1l9),(121),(122) and (162).

Entailment is discussed in (3) and (103).

Computablity is discussed in (34).

2.3. Finite many-valued logics.

'lhe treatises in this current are classfied to the following subjects;3-valued logics, desig-

nated value (super designated value), representation of matrix type (sequent type), relations

among logics, Lukasiewicz algebra, mechanical proof and the rest. 3-valued logics are discussed in the following papers; (7), (13), (39), (49), (67), (68),

(70), (71), (74), (76), (92), (9S), (149), (150), (153), (IS5), (164). Thesie discuss various

systems of 3-valued logics.

Designated value (super designated value) is discussed in the following papers; (44), (60),

(106), (107), (125), (132), (165). In m-vaiued propositional calculus, A. Rose calls the values

1,...s "super designated", and the values s+1,...t "designated" (lgs<t<m). Then a for-mula is called a tautology if either it takesa super designated value at least one or it always

takes designated values.

Representation of matrix type (sequent type) is discussed in (1 29), (130) and (1 59).

Relations among logics are discussed in (63) and (137).

Lukasiewicz algebra and Mechanical proof are discussed in (58) and (9 1), respectively.

2.4. Infmite many-valued logics.

Some systems, which are Lukasiewiczian logic, threshold logic, 1imited logic and continu-ous logic etc., are considered in infinite many-valued logics. (19), (151) and (152) discuss one

of the different effects between many-valued logics and infinite many-valued logics with respect

to antmomles. The treatises in this current are classfied to the following subjects; formalization, desig-

nated value, Lukasiewiczian logic (algebra), threshold logic ([O, 1] -valued logic), limited logic,

continuous logic and the rest.

Formalization is discussed in the following papers; (4)<6), (11), (17), (43), (51), (64),(1 20), (1 23), (131), (136).

Designated value is discussed in (12) and (26).

Lukasiewiczian logic (algebra) is discussed in the following papers; (75), (t141), (143),(1op), (1 70).

Threshold logic ( [O , 1 ] -valued logic) is discussed in (80)-(87) by A. Nakamura.

Limited logic is discussed in (177).

Continuous logic is discussed in (21).

2.5. The rest.

The treatises in this current are classfied to the following subjects; relations among logics

(algebras), algebras, modality, axiom of comprehension, type theory, e-operator, quantifier,recursively enumerable axiomatization, determination, fragrnent and the rest.

Relations among logics (algebras) are discussed in the following papers; (35), (42), (46),(55), (73), (85), (87), (94), (1 40), (1 43), (154), (1 60), (161), (17l).

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Algebras are discussed in the following papers;(18), (72), (98), (99), (1Ol), (142).

Modality is discussed in (30), (85), (87), (94) and (154),

Axiom of comprehension is discussed in (19), (20) and (29).

Type theory is discussed in (56), (57) and (89).

e-operator is discussed in (78).

Quantifier is discussed in'(104) and (105).

Recursively enumerable axiomatization is discussed in (77) and (134).

Determination is discussed in (53).

Fragment is discussed in (62), (166), (167) and (192).

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