372 
 

 American Scientific Research Journal for Engineering, Technology,  and Sciences  (ASRJETS) 
ISSN (Print) 2313-4410, ISSN (Online) 2313-4402 

© Global Society of Scientific Research and Researchers  
http://asrjetsjournal.org/  

 

𝒏𝒏 −Fold 𝒎𝒎−Valued Logic 

G. Nandasenaa*, L. N. K. de Silvab, K. K. W. A. S. Kumarac 

aLecturer, Department of Mathematics and Philosophy of Engineering, Faculty of Engineering Technology, The 

Open University of Sri Lanka, Nawala, Nugegoda, 10250. Sri Lanka 
bFormer Dean of the Faculty of Natural Science in the University of Kelaniya, 109/1, Railway Avenue, 

Maharagama, 10280. Sri Lanka 
cSenior Lecturer, Department of Mathematics, Faculty of Applied science, University of Sri Jayewardenepura, 

Gangodawilla, Nugegoda 10250. Sri Lanka 
aEmail: nandasena1966@gmail.com 

bEmail: nalink2003@yahoo.com 
cEmail: sarath@sjp.ac.lk 

 

 

Abstract  

In this paper, the basic concepts of two valued logic, many valued logic and catuskoti logic are discussed. We 

comprehensively analyzed the statement in different branches of logic. We define a ‘fold’ and a ‘statement’ in 

catuskoti. We further defined a ‘fold’ in general, a ‘multi valued statement’ and ‘𝑛𝑛 −fold 𝑚𝑚 −  valued logic’ 

(where 𝑛𝑛,𝑚𝑚 are integers greater than or equal to two). We identify that every branch of logic could be expressed 

in the form of 𝑛𝑛 −fold 𝑚𝑚 − valued logic. We explain the problem of drawing a tangent to a curve and Zeon's 

arrow paradox by using the 4 −fold 2 − valued logic. Truth tables for negation, conjunction, disjunction and 

implication in the 4 −fold 2 − valued logic are discussed in this paper. 

Keywords: Catuskoti; Many valued logic; Multi valued statement; n −fold m − valued logic.  

1.  Introduction 

Aristotle introduced the classical two valued logic that is called Aristotelian logic. So far, it has been dominating 

the world. A statement in classical two valued logic is defined as a sentence that can be considered as true or 

false. However, there are sentences with number of truth values greater than two. Using the sentences that have 

truth values other than true and false, many valued logics are defined. Such logics are three valued logic, four 

valued logic,  𝑘𝑘 −valued logic (where 𝑘𝑘 is an integer greater or equal to two) and infinite valued logic. 

------------------------------------------------------------------------ 

* Corresponding author.  

http://asrjetsjournal.org/


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 In Buddhist philosophy, there is another different type of logic called catuskoti or tetra lemma or four corner 

logic or four-fold logic. In early Buddhist logic, it was the standard to assume that for any state of affairs, there 

were four possibilities; that it held, that it did not, both or neither [1]. Within the literature survey, we could not 

find clear definitions of a statement and a fold in Catuskoti. In this paper, we explain some phenomenon, which 

cannot be understood in classical two valued logic by using 4-fold 2-valued logic. This work is limited to a 

discussion of the basic concepts of logics up to the operations of statements in 𝑛𝑛 −fold 𝑚𝑚−valued logic. The 

axiomatizations of logics are not discussed in this paper and the applications are confined to the 4 −fold 

2 −valued logic.   

2.  Logic 

The logics were developed in many cultures, including Judeo Christian, Catholic, Hindu, Islamic and Buddhist. 

A logic is a method of obtaining conclusion from a given set of statements in a system of knowledge. Logic is 

discussed mainly in categories, namely, Aristotelian logic, many valued logic, fuzzy logic and Catuskoti logic. 

Scientists, Mathematicians and Philosophers have been using Aristotelian logic to infer the conclusions in 

systems of knowledge. The logic in Catuskoti, which is a Buddhist epistemological concept, is also applied in 

systems of knowledge in cultures based on Buddhist Philosophy. A logic is a set of concepts, and a set of rules 

involving the concepts, which is used for arguments by human beings. The set of concepts and the set of rules 

are relative to the culture of the human beings. 

3. Preliminary Concepts 

The following definitions of concepts are from various authors as stated in the literature. 

3.1 Two Valued Statement 

A sentence that is capable of being either true or false is defined as a statement or a proposition [2]. It is also 

called declarative sentence [2]. Any statement is either true or false but not both. 

3.2 Aristotelian Logic 

Aristotle identified that every science could be described as a system starting from a finite set of basic rules 

called axioms and undefined terms to derive results as well as in Euclidean Geometry. The basic principles to 

derive the results from the axioms were the same for every science. Aristotelian logic is sometimes defined as 

the science of reasoning [3]. 

3.2.1 Concepts and Fundamental Principles of Aristotelian Logic 

In the following sections the concepts and fundamental principles of Aristotelian logic are discussed.   

3.2.1.1 Subject 



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A subject is typically an individual entity, for example a dog, a table, a woman or a group such as a set of 

numbers, a group of people etc [3]. 

3.2.1.2 Predicate 

A predicate is the property or attribute or mode of existence of the subject [3]. A given subject possesses or does 

not possess a given predicate. The color (property) of the box (subject) is red.  All the women (subject) in the 

world are kind (predicate). 

3.2.2 Fundamental Principles of Predication in Aristotelian Logic 

In the literature, the fundamental principles of predication are stated as follows.  

3.2.2.1 The Law of ‘Identity 

Everything is what it is. In symbolic form, it can be expressed as 𝐴𝐴 is 𝐴𝐴 [3]. This implies that an object called 𝐴𝐴 

exists. Modern formulation of the above law is ∀𝑥𝑥�𝐴𝐴(𝑥𝑥) ⇒ 𝐴𝐴(𝑥𝑥)� [3]. 

3.2.2.2 The Law of ‘Non-Contradiction’ 

It is impossible that for a subject both to be and not to be. A given predicate cannot both belong and not belong 

to a given subject in a given respect at a given time [3]. This implies that contradictions do not exist. That is 𝐴𝐴 

and ¬𝐴𝐴 cannot be at the same time. Modern formulation of the above law is ¬∃𝑥𝑥�𝐴𝐴(𝑥𝑥) ∧ ¬𝐴𝐴(𝑥𝑥)�. 

3.2.2.3 The Law of ‘Either-Or’ 

Everything must either be or not be. A given predicate either belongs or does not belong to a given subject in 

given respect at a given time [3]. That is either 𝐴𝐴or ¬𝐴𝐴 be at the same time. This law is also called The Law of 

Excluded Middle [3]. Modern formulation of the above law is ∀𝑥𝑥�𝐴𝐴(𝑥𝑥) ∨ ¬𝐴𝐴(𝑥𝑥)� [3].   

3.3 Many Valued Logics  

So far we have been confined to statements (propositions) which are either true or false. We call these 

statements two valued statements, and a truth value was assigned to a statement depending on whether it was 

true or false. However, there may be sentences with more than two truth values. We call these sentences 

𝑛𝑛 −valued statements (propositions) defined on the number of truth values 𝑛𝑛 that the sentence may take. A logic 

that has many valued statements are called many valued logics. Many valued logic is also known as multi-

valued logic or multiple-valued logic.  

3.3.1 Three Valued Logics 

 An 𝑛𝑛 − valued logic with three truth values is called three valued logic. The three values may be defined as 



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true, false and unknown in western logic. These three truth values are denoted by 𝑇𝑇,𝐹𝐹 and 𝐼𝐼. Truth value of a 

3 −valued statement 𝒜𝒜 is denoted by 𝑇𝑇(𝒜𝒜) [4]. 𝑚𝑚𝑚𝑚𝑥𝑥{𝑇𝑇,𝐹𝐹} = 𝑇𝑇, 𝑚𝑚𝑚𝑚𝑥𝑥{𝑇𝑇, 𝐼𝐼} = 𝑇𝑇, 𝑚𝑚𝑚𝑚𝑥𝑥{𝐼𝐼,𝐹𝐹} = 𝐼𝐼, 𝑚𝑚𝑚𝑚𝑛𝑛{𝑇𝑇,𝐹𝐹} =

𝐹𝐹, 𝑚𝑚𝑚𝑚𝑛𝑛{𝑇𝑇, 𝐼𝐼} = 𝐼𝐼 and 𝑚𝑚𝑚𝑚𝑛𝑛{𝐼𝐼,𝐹𝐹} = 𝐹𝐹. Let  𝒜𝒜 and ℬ be two 3 − valued statements in a 3 −valued logic. In the 

definition of truth values of conjunction and disjunction are defined by using formulae 

𝑇𝑇(𝒜𝒜 ∧ ℬ) = 𝑚𝑚𝑚𝑚𝑥𝑥{𝑇𝑇(𝒜𝒜),𝑇𝑇(ℬ)} and 𝑇𝑇(𝒜𝒜 ∨ ℬ) = 𝑚𝑚𝑚𝑚𝑛𝑛{𝑇𝑇(𝒜𝒜),𝑇𝑇(ℬ)} respectively [4]. 

3.3.1.1 Kleene Logic (strong)𝑲𝑲𝟑𝟑 and Priest Logic 𝑷𝑷𝟑𝟑 

In the literature, the difference between Kleene logic (strong) 𝑲𝑲𝟑𝟑 and Priest logic 𝑷𝑷𝟑𝟑 was stated as follows. In 

Kleene's logic 𝑲𝑲𝟑𝟑  can be interpreted as being “underdetermined”, being neither true nor false, while in Priest's 

logic 𝑷𝑷𝟑𝟑 can be interpreted as being “overdetermined”, being both true and false. Negation, conjunction, 

disjunction and material implication in Kleene (strong) logic 𝑲𝑲𝟑𝟑 and Priest logic 𝑷𝑷𝟑𝟑 are defined as follows [5]. 

In defining the truth values of  𝒜𝒜 ⇒𝐾𝐾 ℬ and 𝒜𝒜 ⇔𝐾𝐾 ℬ, it is used that the equivalence 𝒜𝒜 ⇒𝐾𝐾 ℬ ≡ ¬𝒜𝒜 ∨ ℬ. 

Table 1: Truth Values of Negation in 𝐾𝐾3 and 𝑃𝑃3 [5] 

𝒜𝒜 ¬𝒜𝒜 

𝑇𝑇 𝐹𝐹 

𝐹𝐹 𝑇𝑇 

𝐼𝐼 𝐼𝐼 

Table 2: Truth Values of Other Operators of 𝐾𝐾3 and 𝑃𝑃3 [5] 

𝒜𝒜 ℬ 𝒜𝒜 ∧ ℬ 𝒜𝒜 ∨ ℬ 𝒜𝒜 ⇒𝐾𝐾 ℬ 𝒜𝒜 ⇔𝐾𝐾 ℬ 

𝑇𝑇 𝑇𝑇 𝑇𝑇 𝑇𝑇 𝑇𝑇 𝑇𝑇 

𝑇𝑇 𝐼𝐼 𝐼𝐼 𝑇𝑇 𝐼𝐼 𝐼𝐼 

𝑇𝑇 𝐹𝐹 𝐹𝐹 𝑇𝑇 𝐹𝐹 𝐹𝐹 

𝐼𝐼 𝑇𝑇 𝐼𝐼 𝑇𝑇 𝑇𝑇 𝐼𝐼 

𝐼𝐼 𝐼𝐼 𝐼𝐼 𝐼𝐼 𝐼𝐼 𝐼𝐼 

𝐼𝐼 𝐹𝐹 𝐹𝐹 𝐼𝐼 𝐼𝐼 𝐼𝐼 

𝐹𝐹 𝑇𝑇 𝐹𝐹 𝑇𝑇 𝑇𝑇 𝐹𝐹 

𝐹𝐹 𝐼𝐼 𝐹𝐹 𝐼𝐼 𝑇𝑇 𝐼𝐼 

𝐹𝐹 𝐹𝐹 𝐹𝐹 𝐹𝐹 𝑇𝑇 𝑇𝑇 

 

3.3.1.2 Bochvar's Internal Three-Valued Logic 

Another three-valued logic is Bochvar's “internal”. It is denoted by 𝐵𝐵3𝐼𝐼 . It is also called Kleene's weak three-

valued logic. Truth tables for negation and bi-conditional in Bochvar's internal three-valued logic are the same 



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as in Kleene logic (strong) 𝐾𝐾3 and Priest logic 𝑃𝑃3. Material implication in Bochvar's internal three-valued logic 

is defined by using the equivalence 𝒜𝒜 ⇒ ℬ ≡ ¬𝒜𝒜 ∨ ℬ [4]. Truth tables for conjunction, disjunction and 

material implication in Bochvar's internal three-valued logic are as follows.  

Table 3: Truth Values of Other Operators in Bochvar's Internal Three-Valued Logic [4] 

𝒜𝒜 ℬ 𝒜𝒜 ∧+ ℬ 𝒜𝒜 ∨+ ℬ 𝒜𝒜 ⇒+ ℬ 𝒜𝒜 ⇔+ ℬ 

𝑇𝑇 𝑇𝑇 𝑇𝑇 𝑇𝑇 𝑇𝑇 𝑇𝑇 

𝑇𝑇 𝐼𝐼 𝐼𝐼 𝐼𝐼 𝐼𝐼 𝐼𝐼 

𝑇𝑇 𝐹𝐹 𝐹𝐹 𝑇𝑇 𝐹𝐹 𝐹𝐹 

𝐼𝐼 𝑇𝑇 𝐼𝐼 𝐼𝐼 𝐼𝐼 𝐼𝐼 

𝐼𝐼 𝐼𝐼 𝐼𝐼 𝐼𝐼 𝐼𝐼 𝐼𝐼 

𝐼𝐼 𝐹𝐹 𝐼𝐼 𝐼𝐼 𝐼𝐼 𝐼𝐼 

𝐹𝐹 𝑇𝑇 𝐹𝐹 𝑇𝑇 𝑇𝑇 𝐹𝐹 

𝐹𝐹 𝐼𝐼 𝐼𝐼 𝐼𝐼 𝐼𝐼 𝐼𝐼 

𝐹𝐹 𝐹𝐹 𝐹𝐹 𝐹𝐹 𝑇𝑇 𝑇𝑇 

 

3.3.1.3 Lukasiewicz Logic 

The Lukasiewicz logic has the same tables for conjunction, disjunction, and negation as the Kleene logic given 

above since the truth values are defined by the following formulae 𝑇𝑇(𝒜𝒜 ∧ ℬ) = 𝑚𝑚𝑚𝑚𝑛𝑛{𝑇𝑇(𝒜𝒜),𝑇𝑇(ℬ)} and 𝑇𝑇(𝒜𝒜 ∨

ℬ) = 𝑚𝑚𝑚𝑚𝑥𝑥{𝑇𝑇(𝒜𝒜),𝑇𝑇(ℬ)} respectively. However, it differs in its definition of material implication. This section 

follows the presentation from Malinowski's chapter of the handbook of the history of logic. 𝒜𝒜 ⇒ ℬ is denoted 

by 𝐼𝐼𝐼𝐼𝑃𝑃𝐿𝐿(𝒜𝒜,ℬ). 

Table 4: Truth Values of 𝐼𝐼𝐼𝐼𝑃𝑃𝐿𝐿(𝒜𝒜,ℬ) [4]. 

𝒜𝒜 ℬ 𝒜𝒜 ⇒ ℬ 

𝑇𝑇 𝑇𝑇 𝑇𝑇 

𝑇𝑇 𝐹𝐹 𝐹𝐹 

𝑇𝑇 𝐼𝐼 𝐼𝐼 

𝐹𝐹 𝑇𝑇 𝑇𝑇 

𝐹𝐹 𝐹𝐹 𝑇𝑇 

𝐹𝐹 𝐼𝐼 𝑇𝑇 

𝐼𝐼 𝑇𝑇 𝑇𝑇 

𝐼𝐼 𝐹𝐹 𝐼𝐼 

𝐼𝐼 𝐼𝐼 𝑇𝑇 

 



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3.3.2 Four Valued Logic 

A logic where statements are assigned four truth values namely true, false, over determined and under 

determined denoted by 𝑇𝑇,𝐹𝐹,𝐵𝐵 and 𝐷𝐷 is defined as a four-valued logic [4]. 

3.3.2.1 Belnap Logic  

Belnap's logic is denoted by 𝐵𝐵4 combines 𝐾𝐾3 and 𝑃𝑃3. Negation, conjunction and disjunction in Belnap's logic  

𝐵𝐵4are defined as follows. Belnap logic has no table for an implication connective [4]. 

Table 5: Truth Values Negation in 𝐵𝐵4 [4]. 

 

 

 

 

Table 6: Truth Values of Other Operators in 𝐵𝐵4 [4]. 

𝒜𝒜 ℬ 𝒜𝒜 ∧+ ℬ 𝒜𝒜 ∨+ ℬ 

𝑇𝑇 𝑇𝑇 𝑇𝑇 𝑇𝑇 

𝑇𝑇 𝐵𝐵 𝐵𝐵 𝑇𝑇 

𝑇𝑇 𝑁𝑁 𝑁𝑁 𝑇𝑇 

𝑇𝑇 𝐹𝐹 𝐹𝐹 𝑇𝑇 

𝐵𝐵 𝑇𝑇 𝐵𝐵 𝑇𝑇 

𝐵𝐵 𝐵𝐵 𝐵𝐵 𝐵𝐵 

𝐵𝐵 𝑁𝑁 𝐹𝐹 𝐹𝐹 

𝐵𝐵 𝐹𝐹 𝐹𝐹 𝐵𝐵 

𝑁𝑁 𝑇𝑇 𝑁𝑁 𝑇𝑇 

𝑁𝑁 𝐵𝐵 𝐹𝐹 𝑇𝑇 

𝑁𝑁 𝑁𝑁 𝑁𝑁 𝑁𝑁 

𝑁𝑁 𝐹𝐹 𝑁𝑁 𝐹𝐹 

𝐹𝐹 𝑇𝑇 𝐹𝐹 𝑇𝑇 

𝐹𝐹 𝑇𝑇 𝐹𝐹 𝐹𝐹 

𝐹𝐹 𝑁𝑁 𝐹𝐹 𝑁𝑁 

𝐹𝐹 𝐹𝐹 𝐹𝐹 𝐹𝐹 

 

𝒜𝒜 ¬𝒜𝒜 

𝑇𝑇 𝐹𝐹 

𝐵𝐵 𝐵𝐵 

𝑁𝑁 𝑁𝑁 

𝐹𝐹 𝑇𝑇 



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3.3.3 Gödel's Many Valued Logics 

In 1932, Gödel defined a family of many-valued logics denoted by 𝐺𝐺𝑘𝑘 with finitely many truth values, 

0,
1

𝑘𝑘 − 1
,

2
𝑘𝑘 − 1

, … … … 
𝑘𝑘 − 2
𝑘𝑘 − 1

, 1. 

For example, 𝐺𝐺3 has the truth values  

0,
1
2

 and 1. 

 and 𝐺𝐺4 has 

0,
1
3

,
2
3

 and 1. 

In a similar manner, he defined a logic with infinitely many truth values, 𝐺𝐺∞ in which the truth values are all the 

real numbers in the interval [0, 1] [4]. The truth values of a statement 𝒜𝒜 in Gödel logics 𝐺𝐺𝑘𝑘 is denoted by 𝑇𝑇(𝒜𝒜). 

3.3.3.1 Truth Values of Conjunction in 𝑮𝑮𝒌𝒌 

Let 𝒜𝒜 and ℬ be any two statements. The truth value of a conjunctive statement is defined as follows 

𝑇𝑇�𝒜𝒜 ∧𝐺𝐺𝑘𝑘 ℬ� = min{𝑇𝑇(𝒜𝒜),𝑇𝑇(ℬ)}, 𝑇𝑇�𝒜𝒜 ∧𝐺𝐺∞ ℬ� = lim𝑘𝑘→∞(min{𝑇𝑇(𝒜𝒜),𝑇𝑇(ℬ)}) [4]. 

3.3.3.2 Truth Values of Disjunction in 𝑮𝑮𝒌𝒌 

Let 𝒜𝒜 and ℬbe any two statements. The truth value of a disjunctive statement is defined as follows  

𝑇𝑇�𝒜𝒜 ∨𝐺𝐺𝑘𝑘 ℬ� = max{𝑇𝑇(𝒜𝒜),𝑇𝑇(ℬ)}, 𝑇𝑇�𝒜𝒜 ∨𝐺𝐺∞ ℬ� = lim𝑘𝑘→∞(max{𝑇𝑇(𝒜𝒜),𝑇𝑇(ℬ)}) [4]. 

3.3.3.3 Negation in 𝑮𝑮𝒌𝒌 

Let 𝒜𝒜 be any statement. The truth value of negation of statement 𝒜𝒜 is defined as follows: 

 𝑇𝑇�¬𝐺𝐺𝑘𝑘𝒜𝒜� = �1 if 𝑇𝑇(𝒜𝒜) = 1
0 if 𝑇𝑇(𝒜𝒜) = 0 [4]. 

3.3.3.4 Implication in 𝑮𝑮𝒌𝒌 

Let 𝒜𝒜 and ℬ be any two statements. The truth value of implication is defined as follows: 

 𝑇𝑇�𝒜𝒜 ⇒𝐺𝐺𝑘𝑘 ℬ� = �1 if 𝑇𝑇(𝒜𝒜) ≤  𝑇𝑇(ℬ)
0 if 𝑇𝑇(𝒜𝒜) > 𝑇𝑇(ℬ)  [4]. 



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3.4 Fuzzy Logic 

A logic with statements considering two values namely true and false, and assigning two values 𝑝𝑝 and 1 − 𝑝𝑝 for 

truth and false respectively, where 0 < 𝑝𝑝 < 1 is defined as Fuzzy logic. There are many infinitely many Fuzzy 

logics, where each Fuzzy logic is a two-valued logic, in which truth value of any statement is a real 

valuebetween zero and one. 

3.5 Catuskoti 

Catuskoti known as tetra lemma in Greek is a peculiar reasoning method that is used to explain some Buddhist 

thoughts by Lord Buddha. Even today, Buddhist disciples in Asian countries apply Catuskoti to explain the 

uncertainty of the human life. Catuskoti is a Pali ward. Catus represents four and the meaning of the koti is 

conner or fold. 

3.5.1 Statement in Catuskoti 

We may define a statement in Catuskoti as follows. Let 𝒜𝒜 be a two-valued statement and ¬𝒜𝒜 be the negation of 

𝒜𝒜. If 𝒜𝒜 and ¬𝒜𝒜 can be assigned the same truth value true or false at the same time, then 𝒜𝒜 is defined as a 

statement in Catuskoti.  

3.5.2 Fold in Catuskoti 

In Catuskoti, the truth values of a statement 𝒜𝒜 and its negation ¬𝒜𝒜 can be arranged as the following four 

different ways.  

(a) 𝒜𝒜 is true and ¬𝒜𝒜 is false.  

(b) 𝒜𝒜 is false and ¬𝒜𝒜 is true.  

(c) 𝒜𝒜 is true and ¬𝒜𝒜 is true.  

(d) 𝒜𝒜 is false and ¬𝒜𝒜 is false.   

We may define a corner or a fold in Catuskoti as follows. Any one of the above ways is defined as a corner or a 

fold in Catuskoti. Since there are four different ways regarding any statement 𝒜𝒜 and its negation ¬𝒜𝒜 it is said 

to be Catuskoti. The first two ways correspond to Aristotelian logic and the last two ways are not sensory 

perceptible.  

3.5.3 Logic in Catuskoti 

A logic having two valued statements and including the above four ways is defined as a Catuskoti logic.   

4. Logics with Many Folds 

Let 𝒜𝒜 be a statement in any logical system and its negation be ¬𝒜𝒜. Two valued logic and many valued logics, 

which we have considered so far do not include the cases where 𝒜𝒜 and ¬𝒜𝒜 are both true at the same time, or 𝒜𝒜 



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380 
 

and ¬𝒜𝒜 are both false at the same time. There are no such cases in Aristotelian logic. In this section, we are 

concerned with logics where 𝒜𝒜 and ¬𝒜𝒜 are both true, or 𝒜𝒜 and ¬𝒜𝒜 are both false at the same time. 

4.1 Multi Valued Statement 

We may define multi valued statement as fallows. Let 𝐴𝐴 be two valued or a many valued statement and ¬𝐴𝐴 be 

the negation of 𝐴𝐴. If 𝐴𝐴 and ¬𝐴𝐴 can be assigned the same truth value true or false at the same time then, 𝐴𝐴 is 

defined as a multi valued statement. 

4.2 Fold in General 

Let 𝒜𝒜 be a statement in any logic and ¬𝒜𝒜 be its negation. We define an assignment of possible truth values to 

the statements 𝒜𝒜 and ¬𝒜𝒜 as a fold. The maximum number of folds in an 𝑚𝑚 −valued logic is 𝑚𝑚2. Example, in 

Aristotelian logic there are two truth values so the maximum number of fold is 22. They are given by the 

following table. 

Table 7: Folds of Aristotelian Logic 

𝒜𝒜 ¬𝒜𝒜 

𝑇𝑇 𝐹𝐹 

𝐹𝐹 𝑇𝑇 

𝑇𝑇 𝑇𝑇 

𝐹𝐹 𝐹𝐹 

Example, in 3 − valued logic, there exists three truth values for any statement 𝒜𝒜 so the maximum number of 

folds 3 − logic is 32 = 9. 

Table 8: Folds for a Three-Valued Logic 

𝒜𝒜 ¬𝒜𝒜 

𝑇𝑇 𝑇𝑇 

𝑇𝑇 𝐹𝐹 

𝑇𝑇 𝐼𝐼 

𝐹𝐹 𝑇𝑇 

𝐹𝐹 𝐹𝐹 

𝐹𝐹 𝐼𝐼 

𝐼𝐼 𝑇𝑇 

𝐼𝐼 𝐹𝐹 

𝐼𝐼 𝐼𝐼 

4.4 𝒏𝒏 −Fold 𝒎𝒎−Valued Logic 



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A logic having an 𝑛𝑛 −number of folds, and an 𝑚𝑚 −number of truth values is defined as an  𝑛𝑛 −fold 𝑚𝑚−valued 

logic. Aristotelian logic is a 2 −fold  2 −valued logic. Gödel’s  3 − valued logic (𝐺𝐺3) is a 3 −fold 3 −valued 

logic. 

4.4.1 Logic in Catuskoti is a 𝟒𝟒 −Fold 𝟐𝟐 −Valued Logic 

Let 𝒜𝒜 be statement in a logic and ¬𝐴𝐴 be its negation. Then the logic is called a 2 −fold 2 −valued logic if the 

possible assignments of truth values of 𝐴𝐴 and ¬𝐴𝐴 are as follows. There are only two truth values 𝑇𝑇and 𝐹𝐹 as in 

the case of Aristotelian logic, but they can be assigned in four ways. Hence, it is a 4 −fold 2 −valued logic. The 

truth values of the conjunctive statement 𝒜𝒜 ∧ ℬ and the disjunctive statement 𝒜𝒜 ∨ ℬ are defined by 

𝑇𝑇(𝒜𝒜 ∧ ℬ) = 𝑚𝑚𝑚𝑚𝑛𝑛{𝑇𝑇(𝒜𝒜),𝑇𝑇(ℬ)} and 𝑇𝑇(𝒜𝒜 ∨ ℬ) = 𝑚𝑚𝑚𝑚𝑥𝑥{𝑇𝑇(𝒜𝒜),𝑇𝑇(ℬ)} respectively.The material implication in 

4 −fold 2 −valued logic is defined by using the equivalence 𝒜𝒜 ⇒ ℬ ≡ ¬𝒜𝒜 ∨ ℬ 

Table 9: Truth Values for 4- Fold 2- Valued Logic 

𝒜𝒜 ¬𝒜𝒜 ℬ 𝒜𝒜 ∧ ℬ 𝒜𝒜 ∨ ℬ ¬𝒜𝒜 ∨ ℬ 

𝑇𝑇 𝐹𝐹 𝑇𝑇 𝐹𝐹 𝑇𝑇 𝐹𝐹 

𝐹𝐹 𝑇𝑇 𝐹𝐹 𝐹𝐹 𝑇𝑇 𝑇𝑇 

𝑇𝑇 𝑇𝑇 𝑇𝑇 𝑇𝑇 𝑇𝑇 𝑇𝑇 

𝐹𝐹 𝐹𝐹 𝐹𝐹 𝐹𝐹 𝐹𝐹 𝐹𝐹 

𝑇𝑇 𝐹𝐹 𝐹𝐹 𝐹𝐹 𝑇𝑇 𝐹𝐹 

𝐹𝐹 𝑇𝑇 𝑇𝑇 𝐹𝐹 𝑇𝑇 𝑇𝑇 

𝑇𝑇 𝑇𝑇 𝐹𝐹 𝐹𝐹 𝑇𝑇 𝑇𝑇 

𝐹𝐹 𝐹𝐹 𝑇𝑇 𝐹𝐹 𝐹𝐹 𝑇𝑇 

Taking ℬ to be ¬𝒜𝒜 we obtain the following table. 

Table 10: Truth Table in Catuskoti Related to 𝒜𝒜 and ¬𝒜𝒜 

𝒜𝒜 ¬𝒜𝒜 𝒜𝒜 ∧ ¬𝒜𝒜 𝒜𝒜 ∨ ¬𝒜𝒜 

𝑇𝑇 𝐹𝐹 𝐹𝐹 𝑇𝑇 

𝐹𝐹 𝑇𝑇 𝐹𝐹 𝑇𝑇 

𝑇𝑇 𝑇𝑇 𝑇𝑇 𝑇𝑇 

𝐹𝐹 𝐹𝐹 𝐹𝐹 𝐹𝐹 

From the above table, it is found that the truth value of 𝒜𝒜 ∧ ¬𝒜𝒜 is true in 4 −fold 2 −valued logic whereas it is 

false in Aristotelian logic which is a 2 −fold 2 −valued logic.  𝒜𝒜 ∧ ¬𝒜𝒜 is not a contradiction in 4 −fold 

2 −valued logic. 

 



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4.4.2 Applications of 𝟒𝟒 −Fold 𝟐𝟐 −Valued Logic in Mathematics 

In the following sections, some of the applications of  4 −fold 2 −valued logic are discussed. 

4.4.2.1 Tangents to a Curve 

 

Figure 1: Tangent to a Curve 

Let P and Q be two points on a given curve, where P is a fixed point. The chord PQ and the arc PQ are different 

if P and Q are distinct points. In order for the chord PQ and arc PQ exist P and Q have to be distinct points on 

the curve. Now consider the case when Q approaches gradually to P through the curve. Even if Q is very close 

to P as long as they remain distinct there are two different geometrical objects, one being the chord PQ and the 

other being the arc PQ.  It is said that in the limit as Q tends to P the chord PQ becomes the tangent at P, and the 

arc PQ also in the limit becomes the tangent at P. However, the question arises whether Q coincides with P. If Q 

coincides with P, then there is no chord or arc and hence, we cannot obtain the tangent by considering the case 

where Q coincides with P. In order to obtain a straight line, we have to have two distinct points. In other wards P 

and Q must be distinct points. However close, they may be. On the other hand, we draw a tangent to the curve at 

the point P and not the points Q or neighboring points of P on the curve. Hence, if we consider the tangent as the 

limiting case of chord PQ or arc PQ, then it is necessary that Q has to be distinct from P as well as to be 

coincident with P. Thus, we can say that P and Q are distinct points as well as coincident points (not distinct) 

have to be true. Hence, we cannot draw tangents to a curve in Aristotelian Logic which is 2 −fold 2 −valued 

Logic where the points P and Q are either distinct or coincident but in 4 −fold 2 −valued Logic where P and Q 

can be both distinct and coincident.          

4.4.2.2 Zeon's Arrow Paradox 

A statement that leads to a contradictory conclusion from a sound argument that consists of premises and 

axioms is defined as a paradox [6]. If a paradox is derived in a system of knowledge, we have to reject the 

axioms that affect to the contradiction or to change the logic. Zeon's arrow paradox can be stated as follows. “If 

everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying 

such a space at any moment, the flying arrow is therefore motionless” [7].  “In this paradox, Zeno states that for 

motion to occur, an object must change the position which it occupies. He gives an example of an arrow in 

flight. He states that in any one (duration-less) instant of time, the arrow is neither moving to where it is, nor to 



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where it is not. It cannot move to where it is not, because no time elapses for it to move there; it cannot move to 

where it is, because it is already there. In other words, at every instant of time there is no motion occurring. If 

everything is motionless at every instant, and time is entirely composed of instants, then motion is impossible. 

This paradox starts by dividing time-intervals and not into segments, but into points” [7]. Empirically, the arrow 

moves, but according to Aristotelian logic the conclusion is that arrow cannot move. That means there are no 

solutions for Zeno's paradoxes within the Aristotelian logic. Therefore, the conclusion forces us to change 

axioms or logic or primitive terms. In this case, we change the logic. Instead of Aristotelian logic, we apply 

4 −fold 2 −valued logic. When the arrow moves, it occupies some space that is not constant in the time, the 

occupied space changes from moment to moment. On other hand, when it is at rest, it occupies a constant space 

in the time. While the arrow moves, it occupies some constant space as well as it does not occupy the same 

constant space at the same time. Suppose that the statement S is “the arrow occupies a space”. This implies that 

S is true and ¬𝑆𝑆 is true at the same time. 

5. Conclusions  

In the definition of many valued logics, the results of Aristotelian logic do not contradict. That means if a given 

statement is true, then the negated statement of the given statement is false and vice-versa. Every logic including 

classical two valued, many valued, fuzzy logic and Catuskoti can be expressed in the form of 𝑛𝑛 −Fold 

𝑚𝑚 −Valued logic, where 𝑛𝑛,𝑚𝑚 integers that are greater than are or equal 2. We have identified that in the 

literature, logics other than classical two valued logic have been discussed. These logics are many valued logics, 

infinite valued logics, fuzzy logic and Catuskoti logic (4 −fold 2 −valued logic). We have defined a new logic 

called 𝑛𝑛 −fold 𝑚𝑚 −valued logic such that all the logics are abstracted to the form of 𝑛𝑛 −fold 𝑚𝑚 −valued logic, 

where 𝑛𝑛 and 𝑚𝑚 are natural numbers that are greater than one. We have presented 4 −fold 2 −valued logic. The 

logicians have not gone beyond presenting truth tables. We propose to study the presentation of truth tables in 

detail for 𝑛𝑛 −fold 𝑚𝑚 −valued logic in the future. 

References 

[1] G. Priest “The logic of the Catuskoti” Internet: 2010 [cited 2015 Dec 26]. 1p Available from: 

www.comparativephilosophy.org.  

 [2] Hardegree, “Symbolic Logic” A First Course Mcgraw-Hill College. Internet: 1999 [Dec. 26, 2015] 3p 

Available from: http://courses.umass.edu/phil110-gmh/MAIN/IHome-5.htm 

 [3] G. S. Stephen. “Logic and Mathematics.” Pennsylvania: State University, Department of mathematics 

[Internet]. 2000 [Mar. 26, 2015] 3p Available from: www.personal.psu.edu/t20/papers/philmath.pdf 

 [4] Wikipedia, the free encyclopedia. “Many-valued logic”. Internet: https://en.wikipedia.org/wiki/Many-

valued_logic. Aug. 29, 2016; [Feb. 11, 2016]. 

 [5] liquisearch.com “Many-valued Logic - Examples - Kleene (strong) K3 and Priest logic P3” 2010; 1p 

Internet: www.liquisearch.com/manyvalued_logic/examples/kleene_strong_k3_and_priest_logic_p3.  

http://www.comparativephilosophy.org/
http://courses.umass.edu/phil110-gmh/MAIN/IHome-5.htm
http://www.personal.psu.edu/t20/papers/philmath.pdf
https://en.wikipedia.org/wiki/Many-valued_logic
https://en.wikipedia.org/wiki/Many-valued_logic
http://www.liquisearch.com/manyvalued_logic/examples/kleene_strong_k3_and_priest_logic_p3


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[Aug. 28, 2016]. 

 [6] Wikipedia, the free encyclopedia. “Paradox”. Internet: https://en.wikipedia.org/wiki/Paradox. Mar. 16, 

2016; [Mar. 23, 2017].  

 [7] Wikipedia, the free encyclopedia. “Zeno's paradoxes” Internet:  

https://en.wikipedia.org/wiki/Zeno\%27s_paradoxes. Sep. 22, 2016; [Sep. 27, 2016]. 

https://en.wikipedia.org/wiki/Zeno/%27s_paradoxes