Why Classical Logic (U2 ) and U4 Do Not Explode

Why Classical Logic (U2) and U4 Do Not Explode By Graeme Heald 7h March 2017 Abstract In classical logic the principle of explosion is taken to mean that any statement can be proven from a contradiction. As a consequence in U8 and fuzzy logic, explosion to an infinite list of conclusions occurs for a contradictory premise. It will be shown, however, that classical logic (U2) and U4 do not explode, as irrelevance is impossible in these systems. In particular for classical logic only truth and impossibility are allowed and as a consequence when a contradiction occurs the outcome is restricted to these relevant conclusions only. If this is the case then classical logic and U4 can be safely employed in inference knowledge bases. 1. Introduction The principle of explosion (Latin: ex falso (sequitur) quodlibet (EFQ), “from falsehood, anything follows”, or ex contradictione quodlibet (ECQ), “from contradiction, anything follows”), is the law of classical logic according to which any statement can be proven from a contradiction. That is, once a contradiction has been asserted, any proposition can be inferred from it. In symbolic logic, the principle of explosion can be expressed in the following way AA → B For any statements A and B, if A and not-A are both true, then B is true. An infinite list of outcomes or explosion is said occur when the conclusion is not relevant to the contradictory premises. As it is considered to cause trivialisation of the knowledge base, paraconsistent logic by Priest [1, 2] and relevance logic by Belnap [3], for example, do not contain the principle of explosion. These logic systems are inconsistency tolerant and continue to function after a contradiction occurs. U logic is an extension of classical logic and in U8 the logic state of irrelevance is found. It should be noted that classical logic has always equated irrelevance with falsity. Hence, irrelevance can be given a logic state in U8 while it is hidden in classical logic



2.0 U8 logic system and Explosion U8 is composed of three primary logic sets: truth (T), falsity (F) and neutrality (N). 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.

U = T + F +N /N = T + F

N = /(T +F)

/F = T +N T

F

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.

/T = F + N

Figure 1. U8 Logic 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 [4] for a description of the truth tables for disjunction, conjunction and implication. U8 does explode with a contradiction, denoted , because an irrelevant conclusion is one of 8 possibilities and an irrelevant (N) conclusion can be an infinite number of statements. This can be written as contradiction, or impossibility, implies all logical states:  → T, F, , N, /T, /F, /N, U 1

In U8 an impossible premise validly entails any conclusion. An example of a premise implying a relevant conclusion, „If the earth is round then planets are round‟. If the premise becomes impossible such that „the earth is round and not round‟ then there are 8 possible valid conclusions: i) ii) iii) iv) v) vi) vii) viii)

Planets are round (T) Planets are not round (F) Relativity is correct (N) Planets are round and not round () Planets are round or relativity is correct (/F) Planets are not round or relativity is correct (/T) Planets may be round or not round (/N) Planets may be round or not round or relativity is correct (U)

In particular points iii) v) and vi) above entail explosion as there an infinite number of irrelevant conclusions. 2.1 Fuzzy Logic Fuzzy logic rejects the principle of excluded middle with a range of truth vales between truth and falsity, that allow an infinite number of logic states. Truth equals one and falsity equals zero. For a contradiction there may be an infinite number of logic states. As a consequence, it can be seen that contradiction causes „explosion‟ to a possibly infinite number of logic states. In logical terms: AA → {0, T2, T3, , , Tn-2, Tn-1, 1}. 3.0 U2 Classical Logic and U4 U2 is identical with classical and Boolean logic as it has the same logical assignments [5]. As Boolean or classical logic equates with a reduced logic, U2, assignments based upon U8, become: T = U = /N = /F and F =  = N = /T Where T = truth, F = falsity, N = neutrality or irrelevance,  = impossibility, /F = not false, /T = not-true, /N = not-neutral, U = Universal or uncertainty

F =  = N = /T

T = U = /F = /N

Figure 2. Boolean Classical Logic or U2 It has been claimed for classical logic that when a contradiction occurs, an explosion of possibilities results [2]. In a U2 logic system, the neutral logic state is impossible, denoted N = . As a result, an irrelevant conclusion, due to a contradiction cannot occur so the conclusion according to the principle of explosion must be relevant. In U2, the outcome of a contradiction is limited to the two relevant conclusions: truth and impossibility. As an example: the proposition A: The earth is round. Conclusion B: Planets are round. This is written in implication as: A → B. The contradictory proposition, A is: The earth is not round. A.A has two outcomes

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for B, that is impossible, , and true, T. The two relevant outcomes are:, i) Planets are round and not round ii) Planets are round. Similarly a second example is the implication “if x = 2 then x+1 = 3”. The conclusions produced in the principle of explosion for U2: if x = 2 & x  2 then x+1 = 3 & x+1  3 and if x = 2 & x  2 then x+1 = 3 Points to note from the contradiction in U2 are: i)

The conclusion, B, is relevant to the premise, A

ii)

The principle of explosion leads to two outcomes, impossibility and truth only.

iii)

Explosion does not occur as the conclusion is relevant to the premise.

In logical terms: A.A → {, T}. In U2, a contradiction such as, „This colour is green and this colour is not green‟ does not imply the irrelevant and false conclusion, „The moon is made of green cheese‟. Thus, explosion is avoided. It should also be noted that U4 does not explode as irrelevance is impossible in this logic [6].

=N U = /N F = /T

T = /F

Figure 3. U4 Logic 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 = ) while the universal state equals not neutral (U = /N). Please see the diagram in Figure 3 for a description of the logic system. The U4 logic system displays the conditions for classical logic, particularly the case for which N = , neutrality is impossible. In U4 an impossible antecedent validly entails any relevant conclusion. This can be written as impossibility implies all relevant logical states:  → {T, F, , U}. An example, „If the earth is round then planets are round‟. However, an impossible premise, „the earth is round and not round‟ provides four possible valid conclusions: i) ii) iii) iv)

Planets are round Planets are not round Planets are round and not round Planets may be round or not round

It should be noted that the principle of explosion, A.A→ B, has been described in the previous example with the premise „the earth is round and not round‟. In U4 the „explosion‟ of ex-contradictione quod libet is limited to the four conclusions. Accordingly, explosion does not occur in U4.

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4. Conclusion U8 and fuzzy logic are examples of logic systems that explode in the sense given by “Ex contradiction quodlibet”. These systems explode to an infinite list of irrelevant conclusions given a contradictory premise. It has been shown that U2 does not explode in the case of contradiction due to the fact that irrelevance is impossible in this system. Explosion for classical logic, U2, is restricted to two relevant cases, that is, true and impossible conclusions. Similarly, U4 does not explode as irrelevance is impossible and premises must be relevant to conclusions. It is suggested that because neutrality is a hidden logical variable in Boolean and classical logic, an oversight has occurred. Neutrality, or irrelevance, is revealed only in the U8 logic system. If classical logic does not explode then it can be safely employed in inference knowledge bases. U4 also offers a many valued logic with advances over binary classical logic. References [1] G. Priest, “The Logic of Paradox”, Journal of Philosophical Logic, 8, 1, 1979, pp 219-241. [2]“Paraconsistent Logic”, Stanford Encyclopaedia https://stanford.library.sydney.edu.au/archives/spr2013/entries/logic-paraconsistent

of

Philosophy,

[3] N. Belnap “A Useful four-valued logic”, Modern Uses of Multiple Valued Logic, pp 8-37, 1977. [4] G. Heald “An Outline for a Universal Logic System”, Proceedings of the 7th Mediterranean Conference on Control and Automation (MED99) Haifa, Israel - June 28-30, 1999. [5] D. Kaye, Boolean Systems, Longman, London, 1968. [6] G.Heald “Is U4 an advance over classical logic?”, Research Gate, 29th June 2016

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