Does Mathematics Need Something other than Logic?

Katuzi ONO

Mathematical Institute, Nagoya University

Allow me to reason freely in the primitive logic. Then, I will show you how to develop mathematical theories without assuming anything in it.

Introduction "Does mathematics really need something other than logic?" This is a

question I have asked myself over and over again. Something in my mind has tempted me very strongly to answer "No. Mathematics needs nothing other than logic. If mathematics at present needs something other than logic, mathematics in the ideal form should be established purely logically." Most people would agree to the answer idealistically. Anyone would like to establish mathematics purely logically, if he could. But, how can he know that we can establish mathematics purely logically without being able to do so? I have always believed mathematics is absolutely applicable. If something is concluded mathematically, I believe it absolutely. If there arises something which contradicts a mathematical conclusion, I would only seek after illusional perceptions of the matter or mis-reasonings leading to the mathematical conclusion. I have to admit that illusional perceptions or mis-reasonings are really possible, but I can not even imagine such a situation that some correct perceptions contradict correct mathematical conclusions. Why can we believe in mathematics so absolutely? The reason seems to be that mathematics can be established purely logically. I do not know whether all the theories of mathematics are really established purely logically, but some thing in my mind forces me to say that all mathematical theories should be es tablished purely logically. In reality, however, we live in the tradition of mathematics, in the long lived tradition from the era of Euclid, which is to develop all mathematical theories starting from respective systems of axioms. I do not assert that we have really

developed every mathematical theory starting from a certain system of its axioms , but I must admit that, if not, we are strongly tempted to try so . In the good old days, mathematicians could believe that axioms are self-evi dent truths. Nowadays, we can not agree even that axioms should be self-evident , nor that they could be so. However, if axioms are not self-evident , how can we believe so confidently in the absolute applicability of mathematics?

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Anyway, axioms are something which are not purely logical. If axioms are really necessary for developing mathematical theories, can we still say that mathematics needs nothing other than logic? If we stick to the axiomatic formali zation of mathematics, I think, we can not assert that mathematics needs nothing other than logic. The axiomatic approach has been really effective for clearly disclosing the logical structure of mathematical theories. However, it has not been the sole approach in the logical study of mathematics. In fact, there has been once an epoch when mathematicians seem to have been trying (successfully or not) to construct fundamental mathematical notions such as "real numbers", exclusively on a logical basis. I am thinking that set theory in the beginning was going to be developed in such an atmosphere. Someone seemed to believe that we needed natural numbers even at the beginning of mathematical theories but we can con struct all the others purely logically. Some mathematicians, however, seem to have been trying to construct even the natural numbers "set-theoretically", this perhaps being synonymous with "logically". I think that the category-theoretical approach of these days must be on the same line. A typical example lying along this line in traditional mathematics must be the definition of real numbers by cuts of all rational numbers. We need not regard real numbers always as cuts of all rational numbers, but cuts of all rational numbers really behave just as real numbers. This is why real numbers can be represented nicely by cuts of all rational numbers.

A similar situation arises when we introduce all logical constants in terms of "implication" (notation:"•¨","_??_•¨_??_ is read "_??_ implies _??_") , "universal quanti fication" (notation: "( )", "(ƒÔ)_??_(ƒÔ)" is read "For every ƒÔ,_??_(ƒÔ) holds"), and "nega tion" (notation:"_??_","_??__??_is read "Not _??_"). In classical logic, for instance, "_??_ or _??_" (notation' "_??__??__??_") can be paraphrased as "the denial of W implies 3"

(in notation:"_??__??__??__??_") and "There exists an x which satisfies _??_(ƒÔ)" (notation: "(_??_ƒÔ)_??_(ƒÔ)")can be paraphrased as "It does not hold that _??_(ƒÔ) does not hold for every ƒÔ" (in notation: "- _??_(ƒÔ)_??___??_(ƒÔ)"). We need not regard existence notion always as being defined in terms of universal quantification and negation, but the existence notion thus defined behaves just as the real existence notion in classical logic.

Therefore, the existence notion can be defined nicely in this way in classical logic. These two examples are really similar in some respects. They differ, however, in the respect that, in the former example, mathematical notions only are paraphrased in a system without the notions but, in the latter example, logical constants are paraphrased in a system without the notions. In constructive theories of mathematics, some mathematical notions are introduced, but we keep usually logical constants unchanged. In our study of logical structure of mathematical theories, we are also in

-94- No. 3 Does Mathematics Need Something other than Logic ? 9 terested in mathematical theories standing on the intuitionistic logic, the positive logic, or furthermore, other various logics. In these cases, it seems impossible to keep every logical constant unchanged. In fact, I have been able to develop every finitely axiomatizable formal theory in a very simple logic called primitive logic, which has been introduced in my former papers [2] and [3]. Technical treatment of the matter has been described in details in my former papers [5] and [6]. In the present paper, I will describe our study on the same subject, disclosing my schedules, my failures, and my successes frankly.

(1) Reduction of a Formal Theory to Another by Paraphrasing Every Word of the Former in the Latter To speak of the development of one formal theory in another formal theory or to speak of reduction of one formal theory to another formal theory clearly, I would like to study at first how to define these notions. In mathematics, a typical example of reduction of a formal theory to another would be the reduction of geometry to analysis by introducing coordinates. For reducing plane geometry to analysis, any point is paraphrased by its coordinates, i.e. a pair of real numbers, a straight line is paraphrased as a condition expressed by a linear equation with respect to two variables, and "a straight line passes through a point" is paraphrased as "the coordinates of the point satisfies the equation expressing the straight line" . For plane geometry, "point", "straight line", and "a straight line passes through a point" are basic notions in the sense that other plane geometrical notions can be defined in terms of these basic notions. More generally, every formal theory has a system of its basic notions, in terms of which all the other notions of the theory can be defined. Accordingly, if we give the translations of all the basic notions of a formal theory into another, we would have a complete translation of the former theory into the latter. When every proposition holds in the former theory if and only if its translation holds in the latter theory, the former theory can be duly called "reduced to the latter theory." Reductions in mathematics are usually reductions of this kind. Such a kind of reductions rely on the paraphrasing of their basic notions. These basic notions of a theory are usually called the primitive notions of the formal theory. Any primitive notion of a formal theory is usually denoted by a technical term (maybe a word or a phrase) of the theory, and it is denoted by a special predicate symbol when the theory is formalized. If it is denoted by a phrase, the phrase is translated as a whole as if it were a single word. So, the paraphrasing is essenti ally something like a word-for-word translation. Going more into details, any formal theory has a limited number of its primitive notions. Any other notion in the formal theory is paraphrased in terms of these primitive notions. The paraphrasing is usually called the definition of the

- 95- 10 K. ONO Vol. 3 notion in the theory. Anyway, any sentence in a formal theory can be expressed in principle in terms of its primitive notions (practicable or not). Accordingly, any sentence in a formal theory is completely interpreted in another theory, if every primitive notion of the former theory is paraphrased in terms of admitted notions in the latter theory. Even if this can not be called "word-for-word" in the strict sense, we can easily modulate it as if it is a word-for-word translation. It is enough, 1) to introduce a word in the latter theory corresponding to each primitive notion in the former theory, and 2) instead of paraphrasing every primitive notion of the former theory in the latter, to translate the primitive notion by the corresponding notion just introduced and regard the paraphrasing of the primitive notion of the former in the latter as a paraphrasing of the word in terms of already admitted notions exclusively in the latter theory. If every sentence in a language can be translated literally into another language without any change of meaning and nuance, both languages could be called equivalent in the most strict sense of the word. If only one-way transla tion from the former language to the latter is confirmed to be possible, the latter language is more extensive in its application than the former. In real languages, any literal translation of a sentence would be in principle translated back to the original sentence, but this can not be expected in formal theories. Even in a word-for-word translation of a formal theory into another, it is not certain that every primitive notion of the former is translated into a primitive notion of the latter. For, any literal translation of a formal theory is in principle defined by a literal translation of the primitive notions of the former formal theory to correspond ing words being paraphrased maybe in long phrases in the latter theory. When we would like to translate back the translation of a primitive notion in the same way, we have to retranslate every primitive notion (of the latter theory) occurring in the translation of the primitive notion of the former theory in terms of the primitive notion of the former theory. Accordingly, the phrase obtained by retranslation from the translation of a primitive notion of the former theory may be quite different from the original single primitive notion of the former theory. Any word is usually deemed to denote something, and its translation is deemed to denote the same thing if it is translated literally. In formal theories, however, we can not expect that every word would denote something. Our only concern in formal theories would be in what relation the word stands logically with other words and sentences. Logical relations can be completely given if we know for every sentence whether it is provable or not. Hence, in any formal theory, our concern would concentrate in provability of sentences. Accordingly, any trans lation of a formal theory into another can be duly called literal or faithful when every sentence is provable in the former theory if and only if its translation is provable in the latter. If there is a faithful translation of this kind from a formal

-96- No. 3 Does Mathematics Need Something other than Logic ? 11 theory to another, the former theory is called reducible to the latter. If there is a formal theory, to which many other theories can be reduced, it may be duly called basic. It must be a matter of taste to seek after a basic formal theory or to ask whether there can be a very basic theory. As for myself, I am very much interested in basic theories, (if any exist). I am interested in the formal theory of natural numbers because it is basic to mathematics. I am more interested in set theory because it is even more basic to mathematics. In fact, I have been seeking after a basic theory which would satisfy the following conditions as far as possible A. The theory should be simple and should match with our common sense. B. The class of formal theories that can be reduced to the theory should be very extensive. The class should also include the class of formal theories standing on intermediate logics, i.e. those logics which lie between classical logic and intui tionistic logic including both extremities. C. The faithful translation of any formal theory of the class to the basic theory should be word-for-word. Namely, the translation should be given by translations of only the primitive notions of each formal theory. However, not only had I to give translations of the primitive notions of formal theories but also I had to give translations of the logical constants of the logics they stand on for reducing a vast class of formal theories to a basic theory I had in hand. By admitting translations of this kind, I was successful in reducing a considerably vast class of formal theories to a certain basic theory. In the sequel, I will describe the basic theory I have in hand, I will discuss whether it is really simple and matches nicely with our common sense, I will describe the method of my reductions of formal theories to the basic theory, and I will discuss how far we would have to deviate from our idealistic reduction when dealing with formal theories standing on intermediate predicate logics in general.

(2) The Primitive Logic As a basic theory for all formal theories, I had a very simple logical called primitive logic, which I have introduced in my former papers [2] and [3]. I believe that this logic is simple enough as a basic theory for all formal theories and matches very well with our common sense. At first, let me explain what is meant by the primitive logic.

The primitive logic is a logic having only two logical constants "implication

•¨ and "universal quantification ( )". The inference rules of the primitive logic are : (F) Any proposition "_??_"can be deducedfrom "_??_"itself. (I) Any proposition "_??_"can be deducedfrom the propositions"_??_" and"_??__??__??_"

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("_??_implies _??_"). (I*) The proposition can be deducedfrom thefact that the proposition "_??_"is deduciblefrom the proposition "_??_".

(U) Any proposition of the form "_??_(t)" is deducible from the proposition "(ƒÔ)_??_(ƒÔ)" ("For every ƒÔ , _??_(ƒÔ) holds").

(U*) The proposition "(ƒÔ)_??_(ƒÔ)" is deducible from the fact that the proposition "_??_(t)" is deducible for any arbitrary variable t whatever . I believe, no one would object to my opinion that the primitive logic is simple enough for a basic theory. However, someone object if I assert that my choice of the two logical constants "implication" and "universal quantification" must be unique.

To be fair, I must confess that I have chosen these logical constants some

what arbitrarily for a technical reasons when I began my study in this direc

tion. I must confess that I had also examined the pair of logical constants "implication" and "existential quantification" (notation: "(_??_ )" , "(_??_ƒÔ)_??_(ƒÔ)"means "There is an ƒÔ for which _??_(ƒÔ) holds") together with the pair "implication" and

"universal quantification" . I found in some respect that the former pair seemed

to show more difficulties in reducing many logical systems to a logic having the

pair as its logical constants. I was not sure about the matter, but I had to make a choice anyway. So, I observed the matter from another point of view in asking

which pair would better match with our common sense. I have always believed that mathematics is concerned with the world of possibility rather than the world of reality. Anything can be regarded as possible if there is no fear of implying some contradictions. Possibility is denied only by deducing a contradiction. Deducibility is denoted in propositions by the logical constant "implication". I can not even imagine a logic which does not have "implication" as its logical constant. I have also always believed that our collective scientific mind is perpetually seeking after understanding of our real world by laws. To describe understanding by laws such words as "imply", "all", "every," etc. seem to be indispensable. These words can be described formally by the logical constants "implication" and "universal quantification" .

I know that this does not offer a complete apology for preferring "universal

quantification" to "existential quantification". Especially, in classical logic, "The law _??_(ƒÔ) holds for ev eryƒÔ " can be paraphrased by "There is no exceptional

x for the law _??_(ƒÔ)". In intuitionistic logic, however, we can not paraphrase "universal quantification" in this way in t erms of "existential quantification" and "negation ." Since I have been seeking after a theory which is valid as a basic

theory of classical logic as well as for intuitionistic logic , I have had to proceed anyway in examining whether the primitive logic having the logical constants

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"implication" and "universal quantification" could be really a basic theory as

I had been dreaming.

(3) How to Reduce Classical Logic as well as Intuitionistic Logic to the Primitive Logic

My first problem was to examine whether we could really reduce the classical

predicate logic to the primitive logic. If the primitive logic had another logical constant "negation", this problem would be rather trivial. Namely, classical

logic is a logic for usual mathematics which is concerned with the world of pos

sibility. So, every proposition can be regarded as equivalent to its double denial.

Even in the intuitionistic logic, the denial of any proposition has the property that

its double denial is logically equivalent to itself. Also, logical constants other

than "implication", "universal quantification", and "negation" can be defined in

terms of these logical constants. I have already pointed this out for "disjunction"

(meaning "or") and "existential quantification" in the introduction. For "conjunction" (notation: "_??_" . "_??_•È_??_" means "_??_ and _??_"), "_??_ and _??_" can be

paraphrased in terms of "implication" and "negation by "it does not hold that W implies the denial of _??_" (in notation:"_??_(_??_•¨_??__??_)"). As is well known, the logical

constant "negation" can be replaced by the propositional constant "contradic

tion" (notation "_??_"). Namely, "_??_ does not hold" means "_??_ implies a contradic

tion". But, what is a contradiction formally and intrinsically ? Contradic

tion is not a trivial matter in logical study. Throughout my recent study, I have been concentrating my attention on contradiction, my opinion tending to the position that there is something near the center of our logical activity which is very important and closely connected with contradiction. I call this "something" taboo. Taboo is a proposition for classical logic which has a close connection with usual contradictory propositions. For, any contradictory proposition is regarded as something which can imply every proposi tion, and taboo here is regarded as something which can imply every proposition of a class of propositions we are going to define. For developing the intuitionistic logic by means of a taboo, however, I have had to regard it as a unary predicate which can imply every proposition of a class of propositions we are going to define by taking the value of its variable suitably. Taboo notion in this case deviates considerably from the usual contradiction notion. If we have "contradiction _??_" as a proposition constant, the denial of any pro position "_??_"is paraphrased by "_??_implies _??_"(in notation: so the double denial of "_??_"is paraphrased by "The proposition '_??_implies _??_' imples _??_."(in notation: Because propositions we deal with in classical logic are equivalent to their double denials, we can well expect to have a system similar to classical logic if we deal with such propositions "_??_"only that are equivalent to

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their double denials"(_??_•¨_??_)•¨_??_"

In the primitive logic, however, it is not necessary that the proposition

constant "_??_" has the special quality of contradiction. On the contrary, it is

enough for developing classical logic that "_??_" is a proposition symbol distinct

from all other proposition symbols. So, let us take a new proposition symbol T,

and let us call the proposition "The proposition "_??_ implies T' implies T" (in

notation: "(_??_•¨T)•¨T") the T-closure of the proposition "_??_ (notation: "_??_T").

Let us further call any proposition "_??_ T-closed if and only if "_??_" is equivalent to

its T-closure "_??T_". Then, the world of T-closed propositions shows a great deal

of similarity to the world of possible propositions. Hence, we might have a model

of classical logic in the primitive logic by defining logical constants other than "implication" and "universal quantification" suitably . After a pretty long examination, we can see that this is really the case. Here, I will introduce our definitions of logical constants only. Allow me to be a little technical in defining these logical constants.

Namely, "_??__??__??_" ("_??_ and _??_") is defined by "(_??_•¨(_??_•¨T""_??__??__??_" ("_??_

or _??_") is defined by "(_??_•¨_??_")•¨_??_","__??_"("Not _??_") is defined by "_??_•¨T", and "(_??_ƒÔ)_??_(ƒÔ)"("There is an ƒÔ for which _??_(ƒÔ) holds") is defined by "(ƒÔ)(_??_(ƒÔ)•¨T)•¨T" .

Detailed discussion about this matter can be found in my former papers [3] and [5].

The essential problem of reduction of logics to the primitive logic arises when we

try to reduce intuitionistic logic to the primitive logic. In intuitionistic logic,

we do not know how to express other logical constants than "implication" and "universal quantification" in terms of these logical constants . After a long

struggle, I could define a series of logical constants with respect to a certain

predicate T(.) which resembles the series of logical constants "conjunction", "disjunction" "negation" , and "existential quantification", respectively. So, I will call the newly defined series of logical constants T-conjunction, T-disjunction,

T-negation, T-existential-quantification and denote them by "•È","_??_", "_??_", and "(_??_)" respectively . Corresponding to the T-closure "AT" of a proposition "A" r with respect to a proposition "T", I define also the T-closure "AT" of a proposi

tion "A" with respect to a predicate "T". Namely, I call the proposition "For

every x, the proposition `A implies T(ƒÔ)' implies T(ƒÔ)" (in notation: "(ƒÔ)((A•¨T(ƒÔ))

•¨ T(ƒÔ))") the T-closure of the proposition "A" (notation: "AT"), and I call any

proposition "A" T-closed if and only if "A" is equivalent to its T-closure "AT". The series of logical constants "T-conjunction", "T-disjunction", "T-negation", and "T

existential-quantification" resembles the series of logical constants "conjunction", "disjunction" , "negation", and "existential quantification," respectively, because the logical constants of the former series behave in the world of T-closed proposi

tions just as the respective logical constants in the latter series, assuming that "T"

is a new predicate symbol.

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To be exact, it would be better to define here "T-conjunction •È", "T-dis

junction junction •É ", "T-negation _??_," and "T-existential-quantification (_??_)." Allow me

to be a little technical here in defining these notions. Namely, "_??_•È_??_" is defined by

(ƒÔ)((_??_•¨(_??_•¨T(ƒÔ)))•¨T(ƒÔ))," "_??_•É_??_"is defined by "(ƒÔ)((_??_•¨T(ƒÔ))•¨,((_??_•¨T(ƒÔ))•¨ T(ƒÔ))),""_??_" is defined by "_??_•¨(ƒÔ)T(ƒÔ)," and "(_??__??_)_??_(_??_)"is defined by "(ƒÔ)((_??_)(_??_(_??_)

•¨ T(ƒÔ))•¨T(ƒÔ))." If we take the proposition "T" as the proposition constant "A" in the case of classical logic, the T-closure "AT" of any proposition "A" turns out to be logally equivalent to the original proposition "A" and the series of logical constants defined with respect to "T" turns out to be equivalent to the series of the corresponding logical constants of classical logic. In the case of intuitionistic logic, however, the situation turns out to be quite different. We can not prove any equivalence of this kind in the intuitionistic case. I could prove that the newly defined logical constants behave in the primitive logic just as the corresponding logical constants in intuitionistic logic. Unfor tunately, I have never been able to give a clean cut intrinsic explanation for the question why such a nice situation arises in the world of T-closedpropositions. The T-closure operation for a proposition "T" seems to be something like a sieve, and T-closure operation for a predicate"T" seems to be something like a finer sieve which arranges the whole world of propositions in order depending on T. Speaking frankly, I have always been confident for that T-closure operations must supply a decisive weapon for reducing logics to the primitive logic. Speaking more frankly, I am expecting that T-closure operations will supply a powerful weapon for various studies of logic (for example, the study of truth-value theory).

(4) How to Reduce Formal Theories Standing on Classical Logic or In tuitionistic Logic to the Primitive Logic The theory described in the preceding section leads us to a remarkable result. The T-closures of propositions with respect to a proposition "T" or a predicate "T" play an essential role in the theory, as can be easily noticed. The predicate can stand for any proposition as well as any predicate because propositions can be regarded as a special case of predicates "T", for which any sentence of the form "T(_??_)"is independent of the variable u. So, without loss of generality, we can regard "T" always as a predicate. In place of T, we will take various predicates expressed by sentences, long or short and possibly containing free variables. By taking a predicate "T" in place of the predicate symbol T in the theory described in the preceding section, do we always have a system which can be duly called a formal theory ? Yes. In the system thus obtained all the inference rules of intuitionistic logic hold, so the system can be duly called a formal theory

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" 16 K. ONO Vol. 3 standing on intuitionistic logic. If "T" is a proposition, all the inference rules of classical logic hold in the system, so the system can be duly called a formal theory standing on classical logic. Now, we ask the converse question: Can every formal theory standing on classical logic or intuitionistic logic be reduced to the primitive logic by a certain predicate "T" in the way describedabove ? At present, I can answer this question in the affirmative only to a certain extent. Usual mathematical theories can be formulated as theories standing on classical logic and starting from respective axiom systems. Some of them are finitely axiomatizable, in other words, each of these theories can start from a finite number of axioms. We can answer the question in the affirmative at least for such theories standing on classical logic or on intuitionistic logic that are finitely axiomatizable. This conclusion is really powerful, because the most powerful mathematical theories such as set theory, are formal theories of this kind. Johansson introduced a logic called minimal logic, which does not assume that contradiction implies everything. We can alto introduce logics called positive logicswhich do not have negation notion. We may assume or not assume Peirce's rule "If any proposition 'A' is implied by the propositionthat 'A' implies any other proposition 'B', then the proposition 'A' holds" for these logics. We can answer our question in the affirmative for every finitely axiomatizable formal theory standing on any one of these logics. Now, let us describe a real aspect of reduction of a finitely axiomatizable formal

theory standing on classical logic to the primitive logic. In fact, this is really a

simple case, but also most important as I have already mentioned. Let _??_1, .... _??_,

be the whole system of any finitely axiomatizable formal theory ƒ° standing on

classical logic. Let _??_ be the conjunction of all _??_1's (in notation: "_??_1•È....•È _??_n".

Read "All the propositions _??_1,....,_??_n hold"). Then, _??_ can be regarded as a sole

axiom of the formal theory ƒ°. Hence, the denial of the proposition "_??_"(in nota

tion : "_??__??_") can be regarded as the sole taboo of the system E. However, "•¨_??_"

can never be a proposition of the primitive logic, because the primitive logic does not

have the negation notion. Accordingly, I can not replace the symbol T by this

proposition. To meet the situation, I take up a new proposition symbol T which does not occur in the proposition "_??_", and translate "_??_" word-for-word with

respect to the new proposition symbol T by the method described in the preceding

section.

The word-for-word translation of the proposition "_??_" can be duly called

the sole taboo of the formal theory ƒ° interpreted in the primitive logic. So,

I will take up the translation of "_??_" with respect to the proposition symbol T

as new taboo proposition for the formal theory ƒ°. I will denote the taboo proposi

tion by "_??_", and I will define a new word-for-word translation with respect to the

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new taboo proposition "_??_" by the method described in the preceding section. Now,

we can prove that this reduces the formal theory ƒ° to the primitive logic. In

other words, any proposition "_??_" is provable in the formal system ƒ° if and only

if the word-for-word translation of the proposition "_??_" with respect to the taboo

proposition "_??_" is provable in the primitive logic. We can extend this result also to the intuitionistic case. According to our conclusion, formal theories of certain kinds including the most powerful mathe matical theories can be reduced to an exceedingly simple logic, the primitive logic, by an exceedingly agreeable translation, the word-for-word translation with respect to a taboo predicate. This seems to show how important the primitive logic is. Our proof of this conclusion can not be called simple even in the classical case. Detailed proof is given in my papers [4] and [6]. (The general theory is given in [6], my proof in [4] needs some correction for intuitionistic case.)

(5) Unsatisfactory Results and Future Expectations Some formal systems are formulated as starting from systems of axiom schemes standing on classical logic or intuitionistic logic. Some others are formal theories standing on intermediate logics. Every intermediate logic can be regarded as a sub-logic of a fortified logic of the minimal logic without Peirce's rule. Any logic is called here a fortified logic of another logic if and only if every proposition is meaningful in the former whenever it is meaningful in the latter and every provable proposition of the latter is also provable in the former. Any logic is called a sub-logicof another logic if and only if any proposition is meaningful in the latter whenever it is so in the former and moreover any proposition of the former is provable in the former if and only if it is provable in the latter. To introduce any fortified logic of the minimal logic, we usually need axiom schemes. Axiom schemes are something which can be expressed finitely in ordinary languages but which can not be expected to be expressible finitely in formal theories. For such kind of formal theories, each being introduced by axiom schemes, we can only give a class _??_ of word-for-word translations for each of them such that any proposition is provable in the formal theory if and only if there is a translation in the class _??_for which the translation of the proposition is provable in the primitive logic. I believe this means something, but I must admit that this is not perfectly satisfactory. I wish that we can improve this result more agreeably. I do not know whether there are formal theories which can not be expressed exactly by a finite number of sentences of finite lengths. I do not have any clue to reduce such formal theories faithfully in the primitive logic, each by word-for word translations, but I believe, we can well expect that at least any one of such formal theories that can be expressed in ordinary language by a finite number of

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Literature

[1] Johansson, I., Der Minimalkalkiil, ein reduzierter intuitionistischer Formalismus, Compositio Math., 4(1936), 119-136. [2] Ono, K., A certain kind of formal theories, Nagoya Math. J., 25 (1965), 59-86. [3]-,On universal character of the primitive logic, Nagoya Math J., 27 (1966), 331 353. [4]-,Taboo versus axiom, Nagoya Math. J., 28(1966), 113-117. [5]-,Reduction of logics to the primitive logic, J. Math. Soc. Japan, 19-3 (1967), 384-398. [6]-,On formal theories (To appear). [7]-,A pursuit of simple basic system, Ann. Japan Assoc. Phil. Sci.-, 3-1 (1966), 6-11. [8] Umezawa, T., On logics intermediate between intuitionistic and classical predicate logic. J. Symb. Log., 24 (1959), 141-153.

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