From Classical Logic to Fuzzy Logic and Quantum Logic: a General View
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INTERNATIONAL JOURNAL OF COMPUTERS COMMUNICATIONS & CONTROL Special issue on fuzzy logic dedicated to the centenary of the birth of Lotfi A. Zadeh (1921-2017) Online ISSN 1841-9844, ISSN-L 1841-9836, Volume: 16, Issue: 1, Month: February, Year: 2021 Article Number: 4125, https://doi.org/10.15837/ijccc.2021.1.4125 CCC Publications From Classical Logic to Fuzzy Logic and Quantum Logic: A General View S. Nădăban Sorin Nădăban* Aurel Vlaicu University of Arad, Department of Mathematics and Computer Science, Elena Drăgoi 2, RO-310330 Arad, Romania *Corresponding author: [email protected] Abstract The aim of this article is to offer a concise and unitary vision upon the algebraic connec- tions between classical logic and its generalizations, such as fuzzy logic and quantum logic. The mathematical concept which governs any kind of logic is that of lattice. Therefore, the lattices are the basic tools in this presentation. The Hilbert spaces theory is important in the study of quantum logic and it has also been used in the present paper. Keywords: fuzzy logic, quantum logic, orthomodular lattice, Hilbert space, quantum mechan- ics, effect algebras. 1 Aristotelian logic What is logic? Logic is the science that studies thinking. As in the process of thinking there are two factors, information and emotion, we can talk about a domain of human thought called rational thinking, characterized by the fact that the obtained results are out of the factor called emotion. Therefore we can say that logic is the science that studies rational thinking and we will name it rational thinking logic. The first research about logic was made in the Ancient Greece. The results were systematized by the Greek philosopher Aristotle, in a treatise of logic called Organon. For this reason the rational thinking logic is called Aristotelian logic. In this logic a sentence is either true or false and to one true sentence value 1 is associated while to a false sentence value 0 is associated. Because of this classical logic is also called bivalent logic (having two truth values - true or false). The principles that stay at the basis of this logic are: 1. Principle of Identity: each concept is identical to itself A = A. 2. Principle of Noncontradiction: one sentence cannot be both true and false at the same time. 3. Principle of the Excluded Middle: a sentence is either true or false. "There cannot be an intermediate between contradictions, but of one subject we must either affirm or deny any one predicate" (Aristotle’s Metaphysics). https://doi.org/10.15837/ijccc.2021.1.4125 2 4. Principle of Double Negation: A double negation is an affirmative statement, A¯ = A. This principle was stated as a law of propositional calculus by B. Russell and A.N. Whitehead in Principia Mathematica. 5. Principle of Sufficient Reason: states that everything must have a reason or a cause. This principle led, along the years, to many controversies and various interpretations. Starting from this principle we get to modus ponens, the most important rule of inference, which can be summarized as: if P implies Q and P is true, then Q must be also true. In this situation proposition P is called "sufficient proposition" of proposition Q. I have not yet used the term of mathematical logic. The mathematical logic is the result of a process of mathematical modelling of the rational thinking logic elements, a process that started in the 17th century, by the German mathematician and philosopher G. Leibniz and it was continued in the 19th century by the English mathematicians George Boole and Augustus de Morgan and also by many others along the time. In conclusion, Mathematics, through mathematical modelling of rational logic, brought about an essential contribution to the development of logic. On the other hand, the mathematical logic has become a fundamental instrument in the building of the mathematical universe. 2 The lattices of classical logic In 1854, George Boole discovered a connection between the laws of logic and some laws of algebraic calculus. Thus, he introduced the algebra that was named after him, the Boolean algebra. In this section we briefly review the theory of lattices and we will present two important examples: the lattice of all propositions about a universe of discourse and the lattice of all subsets of an universal set, which, as we will see, can be identified. For other results and notions in the theory of general lattices we refer to [3], [10], [14]. Definition 1. A lattice is a 4-tuple (L, ≤, ∧, ∨) such that (L, ≤) is a partial order set (shortly poset) and for all x, y ∈ L there exists a greatest lower bound (meet or infimum) x ∧ y and a least upper bound (join or supremum) x ∨ y. Proposition 2. If (L, ≤, ∧, ∨) is a lattice, then: 1. x ∨ y = y ∨ x ; x ∧ y = y ∧ x (commutativity); 2. x ∨ (y ∨ z) = (x ∨ y) ∨ z ; x ∧ (y ∧ z) = (x ∧ y) ∧ z (associativity); 3. x ∨ (x ∧ y) = x ; x ∧ (x ∨ y) = x (absorption); 4. x ∨ x = x ; x ∧ x = x (idem potency). Definition 3. A bounded lattice is a lattice (L, ≤, ∧, ∨) with two elements 0 ≤ 1, the least and the greatest elements of L. Proposition 4. Let (L, ≤, ∧, ∨, 0, 1) be a bounded lattice. Then: x ∨ 0 = x, x ∧ 0 = 0, x ∨ 1 = 1, x ∧ 1 = 1, (∀)x ∈ L. Definition 5. A lattice (L, ≤, ∧, ∨) is called distributive if x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z), (∀)x, y, z ∈ L. Remark 6. Condition from previous definition is equivalent to: x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z), (∀)x, y, z ∈ L. Definition 7. Let (L, ≤, ∧, ∨, 0, 1) be a bounded lattice. Let x ∈ L. An element x0 ∈ L is called the complement of x if x ∨ x0 = 1 and x ∧ x0 = 0. The lattice (L, ≤, ∧, ∨, 0, 1) is called complemented if (∀)x ∈ L has a complement in L. The lattice (L, ≤, ∧, ∨, 0, 1) is called uniquely complemented if (∀)x ∈ L has a unique complement in L. https://doi.org/10.15837/ijccc.2021.1.4125 3 Theorem 8. If (L, ≤, ∧, ∨, 0, 1) is distributive and complemented then it is uniquely complemented. Definition 9. A bounded lattice (L, ≤, ∧, ∨, 0, 1) is called Boolean (Boolean algebra or Boolean lattice) if it is distributive and complemented. In the theory of Boolean algebra, the complement is usually denoted by ”0”, but we can also meet the symbol ”¬”. In some papers the term negation is used for the complement, but there are different types of negations which are studied in more general context of non-distributive lattices (see [5]). Therefore, in order to exclude any possible confusion, when we use the term "complement" this will have the meaning of Definition 7. As we will see, in the case of fuzzy logic, we do not have a complement but we do have a negation. Theorem 10. Let (L, ≤, ∧, ∨, 0, 1) be a Boolean lattice. Then: 1. (x ∨ y)0 = x0 ∧ y0 ;(x ∧ y)0 = x0 ∨ y0 (De Morgan law); 2. (x0)0 = x (involution); Example 11. Let F be the family of all proposition about an universe of discourse. If x, y ∈ F we will denote "x implies y" by x ≤ y. It is natural to admit that: 1. x implies x; 2. if x implies y and y implies z, then x implies z; 3. if x implies y and y implies x, then x and y are logically equivalent. Thus (F, ≤) is a poset. The conjunction of two propositions x, y is denoted by x ∧ y and it is the true sentence if x and y are both true and it is false otherwise. The disjunction of two propositions x, y is denoted by x ∨ y and it is the true sentence if, at least one of the sentences x and y is true and false otherwise. Thus (F, ≤, ∧, ∨) is a lattice. We note that (F, ≤, ∧, ∨, 0, 1) is a bounded lattice, where 0 is a proposition that is always false (called a contradiction) and 1 is a proposition that is always true (called a tautology). The complement of the proposition x is the proposition x0 which is true if x is false and it is false if x is true. Example 12. Let X be a nonempty set. The power set P(X) (the family of all subsets of X) is a lattice in which the order relation is inclusion and the operations on P(X) are union and intersection. Moreover, (P(X), ≤, ∧, ∨, 0, 1) is a bounded lattice, where the smallest element 0 is the empty set and the largest element 1 is the set X itself. The complement of A is CX (A). In the theory of the lattices the concept of logical implication is a very important one. This concept is associated with an operation over two logical values. In the classical logic x → y means that x ≤ y which is logically equivalent to x0 ∨ y sau cu x = x ∧ y. In the case of non-Boolean lattices we can also have other ways of defining the logical implication. 3 Orthomodular lattices In this section we will present some aspects of orthomodular lattices, because they represent the basic structure in quantum logic. For other results and notions in the theory of general lattices we refer to [3], [10], [14]. Definition 13. A lattice (L, ≤, ∧, ∨) is called modular if: x ≤ y ⇒ x ∨ (z ∧ y) = (x ∨ z) ∧ y, (∀)z ∈ L. Theorem 14. Every distributive lattice is modular. Definition 15. Let (L, ≤, ∧, ∨, 0, 1) be a bounded lattice. A unary operation ⊥: L → L is called orthocomplementation if (∀)x, y ∈ L we have: https://doi.org/10.15837/ijccc.2021.1.4125 4 ⊥ 1.