A Condition for Paradox

A Condition for Paradox 29/March/2017 By Graeme Heald Abstract Despite the denial by classical logic, paradox exists and can be given by the formula, A  /A. A condition for paradox can then be stated, U = T = /T = , or, U = T0 = T1 = ..= Tn-1 =  for any n truth value logic system. The Venn diagram for set theory does not apply in this case, and paradox is a logic system, not a logic state. This paradox condition is distinguished from that of fuzzy logic and para-consistent logic LP. Russell‟s paradox is an example of a paradox.

1. Introduction Classical logic was founded by Aristotle (384 – 322 B.C.) He is credited with the earliest study of formal logic with the Prior Analytics [1], and his conception was the dominant form of western logic until nineteenth century advances in mathematical logic. As it contains true and false values only the structure of classical logic is bivalent. Classical logic is characterised by the three „laws of thought‟: i) The Law of Identity ii) The Law of NonContradiction iii) The Law of Excluded Middle. Please note that the symbol “/” shall represent the complementary or NOT operation. The Law of Identity can be stated as „whatever is, is.‟ that is, for any proposition A: A = A. ii) The Law of Non-Contradiction can be stated as „Nothing can both be and not be‟. For a proposition A: A AND /A = . iii) The Law of Excluded Middle can be stated as „Everything must either be or not be‟. For a proposition A: A OR /A = U. i)

The earliest recorded incidence of paradox has been credited to Epimenides of Knossos, a Cretan, circa 600 BC when he stated that „All Cretans are liars.‟[2]. If he was lying he was telling the truth and if he was telling the truth he was lying. Eubulides of Miletus, possibly in response to Aristotle, was a Megarian stoic philosopher in the fourth century BC who reportedly asked, „A man says that he is lying. Is what he says true or false?‟[3]. This has become known as the liar paradox, as the proposition appears to be both true and false. In chapter 25 of Sophistical Refutations Aristotle attempted to address the liar paradox. He considers a series of cases in which contradictory propositions appear to be held together without qualification [4] showing that

contradictory predicates can be distinguished. For Aristotle, a contradictory sentence is the result of a mistaken equivocation in respect of one of its factors. As a consequence of the principle of non-contradiction, classical logic following Aristotle rejected paradoxical statements. 2. Analysis of the Liar Paradox Since the classical age the liar paradox has come to be seen as a self-referential statement: (A)

„This statement is not-true‟

If (A) is true, then the statement must be true and, at the same time, (A) must be not-true, leading to a contradiction. If (A) is not-true then the statement is nottrue and therefore (A) must be true. Either way (A) is both true and not-true, which is a paradox. A self referential and paradoxical statement occurs when A implies not A and not A implies A: A → /A and /A → A From the statement (A) it can be seen that the liar paradox can be described by propositional formulae as A equals NOT A: A  /A If for the liar paradox truth, T, is non-truth, /T, the equality, T = /T, can also be given. By the principle of non-contradiction T/T = , means that T  T = T =  and similarly, /T /T = /T = . Hence, the equation for the universal set substituting T and /T is U = T + /T = . Because U = T + /T applies to two valued as well as many valued logic, a general condition for paradox exists in which: U = T = /T = 

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3. Paradox in Other Systems of Logic In order to allow paradoxical truth values and produce a non-trivial system, para-consistent logic rejects the principle of explosion and the principle of noncontradiction. According to Priest para-consistent logic introduces the law of non-triviality. The most well-known system of para-consistent logic LP (Logic of Paradox) was first proposed by Argentinian logician F.G. Asenjo in 1966 and later popularized by Priest. The Logic of Paradox (LP or P3) adds a third “undefined” truth value of both true AND false [5]: F = False, T= true, P = both true AND false. Also OR =  and AND =  Table 1. LP Truth Tables A /A F T P P T F

AB F P T

F F F F

P F P P

T F P T

AB F P T F F P T P P P T T T T T

In rejecting the principle of explosion LP truth tables show that modus ponens does not hold. To circumvent this problem a modification to the implication truth table known as RM3 has been introduced [6]. LP preserves most other inference patterns, such as De Morgan‟s law as well as the usual introduction and elimination for negation, conjunction and disjunction. The tautologies of LP are equivalent to classical propositional logic. Fuzzy logic rejects the principle of excluded middle with a range of truth vales between truth and falsity allowing

an infinite number of logic states. Truth equals one and falsity equals zero. Fuzzy logic has been based on the original work of Lukasiewicz who produced a 3-valued logic system with an indeterminate logic state, i, between truth and falsity [7].

4. U Logic and dimensions of logic systems To illustrate the effectiveness of the Venn diagram representation of logic systems, the dimensions of systems can be considered from 3 to 0 dimensions. Universal logic or U8 proposed by the author is a 3 dimensional logic system, while U4 probability logic are 2 dimensional systems. Classical logic and Boolean logic or U2 is a 1 dimensional system. The 0 dimensional logic system will be shown to be paradox. 4.1 U8 Logic System The U8 logic system is composed of three primary logic sets: truth, falsity and neutrality and these constitute the dimension of the logic system [9]. A truth value termed „neutral‟ is establishes between the contrary terms of truth and falsity (T = ⌐F). As neutral is neither true nor false, N = / (T + F), it can be viewed as being irrelevant or meaningless. There are also three secondary sets: not-true (/T = F + N), not-false (/F = T +N) and not-neutral (/N = T + F). Because there are a total of 8 truth values this system can also be referred to as „U8‟. Please see the diagram in Figure 1 for a representation of the Universal logic system.



 To obtain fuzzy logic definitions [8], let A and B be arbitrary assertions: t(A  B) = min { t(A), t(B)) } t(A  B) = max { t(A), t ( B ) } t ( /A ) = 1 - t(A) t(A) = t(B) if A and B are logically equivalent.

U = T + F +N

/N = T + F

/F = T +N

F /T = N + F T N = / (T + F)

For paradox, the negation of truth is equal to truth, T = /T. This occurs in fuzzy logic when truth equals falsity equals 0.5 [8]: T = 1 - /T = ½.

Figure 1. U8 logic system

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In addition, there are the universal and null ( = /U) logical sets with the universal set as the union of all primary logical states (U = T + F + N). Please see [9] for a description of the truth tables for disjunction, conjunction and implication. 4.2 U4 Logic System In a two dimensional logic system falsity equates with not-true (F = /T), and truth with non-falsity (T=/F), and neutrality with impossibility (N =Φ) while the universal state equals not neutral (U = /N). Please see the diagram in Figure 2 for a description of the logic system [10].

and possibility are intimately related, so that probability could be seen as a type of modality. Similarly, probability logic is a two dimensional logic. 4.3 Boolean or U2 logic system The primary logic value truth gives a one dimensional logic system, where it equates with the universal set and non-falsity and not-neutral. While, the null set is

F =  = N = /T

=N T = U = /N = /F U = /N T = /F

F = /T

Figure 3. Boolean Logic or U2

Figure 2. U4 logic As modal syllogisms were considered by Aristotle in the Prior Analytics Chapters 8 to 22, the two dimensional logic system could be considered to be classical [1]. Traditionally, the „alethic modalities‟ include possibility, necessity and impossibility [8]. It is clear that in terms of U8 necessity can correspond with truth (T), impossibility with null () or falsity (F) and possibility with the Universal state (U). Consequently, modal logic can be considered to be a two dimensional logic. The area of any truth value on a Venn diagram can be equated with probability. For example, the area of probability of A union B is equal to equation: P(AB) = P(A) + P(B) – P(AB). Thus, probability can be regarded as an estimate of truth values and probability theory a natural outgrowth of two valued logic – a probability logic. According to the classical interpretation, probability

equated with falsity and neutrality and non-truth. This is Boolean logic [10] and it can be seen to be a classical logic system. Please see the diagram right in Figure 3 for a description of the Boolean logic system. In Boolean logic all of the theorems of classical logic are valid; specifically the laws of thought and modus ponens. For further clarification, please see truth tables of conjunction, disjunction and negation in Boolean logic [11]. Moreover, it could be inferred that Aristotle only intended classical logic to be a one or two dimensional system. In Boolean logic the universal set is equated with truth, and falsity with impossibility. This is regarded as the „all true‟ state. There is no uncertainty in Boolean logic, and logic states are either on or off. This has proven to be particularly useful with digital systems and artificial intelligence.

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4.4 A Paradoxical Logic System A further reduction from the Boolean system of logic would lead to the zero dimensional logic system, U0, in which all truth values are equated. It can be seen that the zero dimensional logic system is equivalent to the general condition for paradox, indicated previously as U = T = /T = . In Venn diagrams the zero dimensional logic system has no primary truth values and no visual representation.

 (R  R)  (R R) The equivalence of contradictory formula is the condition for paradox. Therefore, Russell‟s paradox is a paradox. 6. Conclusion A propositional formula for paradox can be given for any statement A when: A  /A

All truth values are equated with the null state to provide a mathematical condition for paradox: For U8: U = T = F = N =  For U4: U = T = F =  For U2: U = T =  For U0: U =  The condition for paradox can also be generalised for an n dimensional truth value system where T0 to Tn-1 are truth values and U is the universal state and  is the null state: U = T0 = T1 =….= Tn-1 =  Hence, given the general condition of paradox in which all truth values are equated it can be seen why paradox is sometimes referred to as a „truth value glut‟ [6]. Paradox is a logic system and not a logic state as given by LP and so truth tables cannot be defined. This paradox condition is distinguished from LP where paradox is both true AND false, while, in this case they are simply equivalent. Similarly, this condition for paradox does not equal ½, as is the case for fuzzy logic. 5. Russell’s Paradox The philosopher Bertrand Russell discovered the paradox in 1901 related to set theory [12]. There have been questions whether this is a real paradox and so a brief analysis will be given.

A general condition for paradox has been found for any n value many valued logic system: U = T = /T = , or equivalently, U = T0 = T1 =….= Tn-1 =  This condition of paradox has been distinguished from that of fuzzy logic and para-consistent logic LP. It has been shown that paradox is a logic system and not a logic state. Russell‟s paradox is an example of paradox.

References [1] Aristotle, Prior Analytics, Encyclopaedia Britannica Inc, Uni of Chicago, 1952. [2] T. Fowler, The Elements of Deductive Logic, 3rd Ed., Oxford: Clarendon Press, 1869, p 163, retrieved 1Apr 2011. [3] A. Borghini, “Paradoxes of Eubulides”, About.com (New York Times), Retrieved 04-09-2012. [4] D. Bastiras “Eubulides, Aristotle and Chrysippus on the Liar Paradox”, Uni of Adelaide, www.softlab.ntua.gr/~nickie/tmp/pls5/bastiras.pdf, 2016. [5] G. Priest, “The Logic of Paradox”, Journal of Philosophical Logic, 8, 1, 1979, pp 219-241.

Let R be the set of all sets that are not members of themselves. If R is a member of itself then it must satisfy the condition of not being a member of itself. If it is not, then it must satisfy the condition of not being a member of itself, and so it must be a member of itself.

[6] G. Priest, An Introduction to Non-classical Logic, from if to is, Cambridge, second ed, 2008.

In propositional formula this can be given as:

[8] P. Hajek and J. Paris and J. Shepherdson, “The liar paradox and fuzzy logic”, Journal of Symbolic Logic, Vol. 65, No. 1, pp 339-346, Mar 2000.

(R  R) → (R R) (R R) → (R  R) R appears to be a member of itself if and only if it is not a member of itself.

[7] C. Elkan “The Paradoxical Success of Fuzzy Logic”, IEEE discussion, University of California, San Diego, Aug 1994.

[9] G. Heald “An Outline for a Universal Logic System”, Proceedings of the 7th Mediterranean Conference on

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Control and Automation (MED99) Haifa, Israel - June 28-30, 1999. [10] G.Heald “Is U4 an advance over classical logic?”, Research Gate, 29th June 2016 [11] D. Kaye, Boolean Systems, Longman, London, 1968. [12] Stanford Encyclopaedia of Philosophy Russell’s Paradox, https://plato.stanford.edu/entries/russellparadox/

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Definition of keywords Keywords: paradox, neutral, universal, null Paradox: A proposition or statement that is actually selfcontradictory. A general condition for paradox has been given as the equivalence of all logical states: U = T0 = T1 = ..= Tn-1 =  Neutral: As described in U8, neutral, N, is neither true nor false, N = / (T + F), and it can be viewed as being irrelevant or meaningless. Universal: As described in U8, the universal set is the union of all primary logical states, U = T + F + N, or more generally U = T + /T. Null: The null set, , is the empty set. It is the complement of the universal set,  = /U.

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