Material and Strict Implication in Boolean Algebras,

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1 2014 by the Authors; licensee ECSC. This Open Access article is distributed under the terms of the Creative Commons Attribution License ( Use, distribution, and reproduction in any medium is allowed, provided the original work is cited. Issue 2/ Material and Strict Implication in Boolean Algebras, Revisited Enric Trillas 1 and Rudolf Seising 1 1 European Centre for Soft Computing, ECSC, Mieres, Spain Enric Trillas - enric.trillas@softcomputing.es; Rudolf Seising - rudolf.seising@softcomputing.es; Abstract It can be said that Formal Logic begun by studying an idealization of the statements if p, then q, something coming from long ago in both Greek and Scholastic Philosophy. Nevertheless, only in the XX Century it arrived at a stage of formalization once in 1910 Russell introduced and identified the material conditional with the expresion not p or q. In 1918, and from paradoxical conditionals like If the Moon is a cheese, it is a Lyon s face, Lewis critiziced the material conditional and introduced the so-called strict conditional as the modal necessity of the material one. In 1934, Huntington proved that both material and strict implications are coincidental in the setting of Boolean algebras and, hence, that they only can be actually different in algebraic structures weaker than Boolean algebras. Boolean algebras have too much laws for supporting the difference of both conditionals. This paper is nothing else than an algebraic trial to find simple binary Boolean operations able to express Lewis strict implication, and contains a new and simpler proof than that of Huntington, made by identifying possible with non self-contradictory. By the way, it is shown that this proof is only valid in Boolean algebras, but neither in proper De Morgan algebras, nor in proper Ortholattices, with which it still remains open where the two conditional relations are actually different. Submitted: 02/07/14. Accepted and Published: 03/12/14. Page 1 of 12

2 1 Introduction As it can be seen in the web, in the Logic s literature and from long ago, in the history of logic there is a debate on which of the two implications, strict or material, should be considered and where it should be. Megarian school philosophers Diodorus Cronus and Philo the Dialectician (about 300 bc) introduced the logical connective if...then... in their disputes on the truth of conditional statements. Whereas Philo said that a conditional is not true if a correct antecedent has an incorrect consequent, but in all other cases it is true, his teacher Diodorus said that conditionals are true only when the antecedent could never have a non-true conclusion. We do not know which of these views is older but these different views on conditional statements gave rise to the well-known paradoxes of material implication that arrive when we identify our ordinary conditional if A, then B as material conditional because then it is true for every false A and for every true B [10]. In the second half of the 20 th century there was also an intensive discussion on whether already in medieval logic William of Ockham studied various types of inferences and that he differentiated between today so-called strict and material implications when he considered different types of consequences. In 1951 Boehner argued that Ockham s material consequence is our modern material implication [2]. Twenty years later this interpretation was countered by Mullick [9] and Adams [1]. Whether or not both conditionals or implications were early introduced by the old Greeks and studied by Scholastic philosophers, material implication was introduced as the binary operation p m q = p + q (not p or q), by B. Russell in the Principia Mathematica 1, the famous book he jointly wrote with A. N. Whitehead [17]. In this enormous work on the foundations of mathematics the authors considered disjunctions with the first operand negated: p q (not a or b) and interpreted this form of disjunction as an implication: If p is true, p is false, and accordingly the only alternative left by the proposition p q is that q is true. In other words, if p and p q are both true, then q is true. In this sense, the proposition p q will be quoted as stating that p implies q. The idea contained in this propositional function is so important that it requires a symbolism which with direct simplicity represents the proposition as connecting p and q without the intervention of p. But implies 1 However, Gottlob Frege had already introduced this conditional as a so-called Junktor with name implication in his Begriffsschrift (Concept Notation) in 1879 [4]. Page 2 of 12

3 as used here expresses nothing else than the connection between p and q also expressed by the disjunction not-p or q. [17, p. 7] Already in his Survey of Symbolic Logic in 1918 Clarence Irving Lewis introduced an impossibility operator as a new primitive idea [7]. With this operator and the negation he could also define necessity and non-necessity. Later, in the book Symbolic Logic, jointly written with C. H. Langford [8] this was used to establish the system S3 where the sign was used for possibility. However, Strict implication was introduced by Lewis as a relation. He defined this relation in terms of negation, conjunction, and the 1-place (unary) modal operator, for Possibly 2, as follows: For every formula X with a classical bivalent truth value X is used to indicate possibility in Modal logic. X says X is possibly true (or false). Then Lewis defined A B (A strictly implies B) as (A B). 3 Whereas Material implication says that p and not q is wrong means Strict implication that p and not q is impossible, i.e. necessary false. Therefore, Strict implication was understood as the necessity of the material one 4. What seems to be unknown is whether the strict implication can, or cannot, be also expressed by a binary operation, at least in abstract Boolean algebras [6, 12] B = (B,, +, ; ; 0, 1). Both implications accept that true implies true, is true, and that true implies false, is false, but dissent on the truth or falsehood of false implies true commonly considered as a paradox of material implication. That is, if seen as binary operations, both accept the triplets (a, b, implication) of values (1,1,1), and (1,0,0), the discussion can be centered in which one of the triplets (0,1,1) and (0,1,0) can, respectively, be accepted by those implication operations. Notice that such triplets refer to the absolute truth of the antecedent, the consequent, and the implication, once understood as the maximum (1), and the minimum (0) of the Boolean algebra B. When seen as a relation, the problem is if the pair (0, 1) is, or is not, in it. This short algebraic note just tries to find, in Boolean algebras B and without consciously considering anything of modal logic, but just the algebraic skeleton of such algebras, which two variables Boolean expressions or binary operation expressible by connectives [12], a b = α a b + β a b + γ a b + δ a b, (1) 2 With, for Possibly, there is also the modal operator, for Necessarily and in modal logic each can be expressed by the other and by negation: X X and X = X. Hence (A B) = ( ((A B)) = ( A B) 3 For a modern presentation see [3]. 4 Firstly, Lewis proposed that Russell s material implication should be replaced with strict implication. It was the aim of Lewis to determine that a false antecedent can never strictly imply a true consequent. Page 3 of 12

4 with the coefficients α, β, γ, δ in {0, 1} B, can represent each type of implication [13]. We also try to understand the link between the implication represented by these binary operations, and the implication when it is seen as a binary relation in B. For what concerns the relation s view, what is here obtained essentially coincides with what Edward V. Huntington obtained in 1936 [5], that in Boolean algebras the assertion of Russell s material implication is coincidental with Lewis strict implication relation. Nevertheless, here it is not only posed in a much simpler algebraic way than Huntington did, but it is also shown which other Boolean operation of implication (weaker than material implication operation) can also generate Lewis strict implication relation. 2 Specifying material and strict implication operations We will argue along the following lines (Fig. 1): Figure 1: Argumentation line of types of implication. 2.1 From equation (1), with triplet (1, 1, 1), it follows 1 1 = α = 1; Page 4 of 12

5 2.2 From equation (1), with triplet (1, 0, 0), it follows 1 0 = γ = 0, consequently, equation (1) is reduced to: a b = a b + β a b + δ a b, (2) for β, δ {0, 1}. Now a bifurcation corresponding to both implications can be considered. 2.3 Material implication. From equation (2) and the triplet (0, 1, 1), it follows 0 1 = β = 1, and Hence, it is either, a m b = a b + a b + δ a b. δ = 0, and a m1 b = a b + a b = (a + a ) b = b, or, δ = 1, and a m2 b = a b + a b + a b = a b + a (b + b ) = a b + a = (a + a ) +(b + a ) = b + a = a + b. In this case, it is 0 m1 0 = 0, and 0 m2 0 = Strict Implication. From equation (2) and the triplet (0, 1, 0), it follows 0 1 = β = 0, and a s b = a b + δ a b. Hence it is either, δ = 0, and a s1 b = a b, or δ = 1, and a s2 b = a b + a b. In this case it is 0 s1 0 = 0, and 0 s2 0 = 1. Page 5 of 12

6 2.5 It should be pointed out that implication operations are, in the first place, to symbolically represent conditional statements If a, then b in some formal framework. This does not mean that the operation should be functionally expressible like it is in equation (1), and deserves further study. In the second place, they are for symbolically doing forwards or backwards inference following, respectively, the schemes of Modus Ponens and Modus Tollens. Since the four implications, m 1, m 2, s 1, and s 2, verify the Modus Ponens Inequality: a (a m1 b) = a b b, a (a m2 b) = a (a + b) = a a + a b = a b b, a (a s1 b) = a (a b) = (a a) b = a b b, a (a s2 b) = a (a b+a b ) = a (a b)+a (a b ) = (a a) b+(a a ) b = a b b, all of them allow forwards inference provided it is a b 0. Notice that if it were a b = 0, the inequality will hold for any b B, and then much care should be taken for doing forwards inference. The two operations of implication, the material m2 and the strict s2, are contra-symmetrical: b m2 a = (b ) + a = b + a = a + b = a m2 b, b s2 a = b a + (b ) (a ) = b a + b a = a b + a b = a s2 b, and consequently also allow Modus Tollens backwards inference b (a b) = b (b a ) b a a. It is b m1 a = a, and m1 is not contra-symmetrical, and verifies b (a m1 b) = b b = 0. In the same vein, the strict implication s1 is not contra-symmetrical since b s1 a = b a a b = a s1 b, and in addition, b (a s1 b) = b (a b) = b (b a) = (b b) a = 0. Hence, much care should be taken for doing backwards inference with both m1, and s1. Page 6 of 12

7 Remarks. i) Provided strict implication does not decide on the third value (?) in the triplet (0, 1,?), but just rejects? = 1, then from equation (2), it just should be considered the cases β = 0, δ = 0, and β = 0, δ = 1, respectively reproducing s1 and s2. ii) An open possibility for widening the view to more operations, and not considered in this note, consists in allowing coefficients in equation (1) to take all possible values in the Boolean algebra but not only its extremes 0 and 1. iii) If it is, obviously, a m1 b = b a = b m1 a, and a m2 b b m2 a, it is, a s1 b = b s1 a, and a s2 b = b s2 a, that is, both strict implications show the unreasonable property of being commutative operations. A way of exiting from this trouble is by affecting them by contextual and non-commutative parameters ρ i (a, b), i = 1, 2, verifying ρ i (1, 1) = 1, in the forms: a s 1 b = ρ 1 (a, b) (a b), a s 2 b = ρ 2 (a, b) (a b + a b ), with which 0 s 1 0 = 0, and 0 s 2 0 = ρ 2 (0, 0) whose value can be kept equal to one, like with s2, by just taking ρ 2 (0, 0) = 1. This opens a way to consider implications s 1 and s 2 as non-commutative binary operations that here is just pointed out, and deserves further consideration. iv) From a b a b + a = (a + a ) (b + a ) = b + a, and a b + a b a + b, it follows a b a + b, and a b + a b a + b. Thus, both strict implications, s1 and s2, are weaker operations than the material implication m2. Since a b = a + b implies a b = 0, and a b + a b = a + b implies a b, in general the two former inequalities are not equalities. 3 A view on the implication as a relation 3.1 If a b, then b a and b + b = 1 a + b, that is, a + b = 1. Reciprocally, if a + b = 1, then a (a + b) = 1 a = a, and a b = a, or a b. Hence, Page 7 of 12

8 a m2 b = 1 a b, shows that asserting Russell s material implication operation gives nothing else than the order of the Boolean algebra B. 3.2 In his view of implication as a relation [8], C. I. Lewis, defines strict implication by, that, if interpreting Thus, a b (a b ) = a + b is necessary, p is necessary by not possible not p, possible by non self-contradictory, a is contradictory with b, by a b a b = 0, and with (a + b) = a b, it is obtained, a b a b (a b ) = a + b, since not(non self-contradictory not (a b ) ) is equivalent to not(non self-contradictory a b ). = {(a, b) B B ; a b a + b}. Theorem 3.1. In all lattices with infimum operation, supremum operation +, and an order-reversing unary operation,. Proof. Since a b b a a b b a a + b, it is. Of course, Theorem 3.1 holds in all Ortho-lattices and De Morgan algebras, in particular it holds in Orthomodular lattices and Boolean algebras. In addition, notice, that in the former proof no role is fully played by the lattice s character of the two operations and +. Consequently, Theorem 3.1 also holds in structures weaker than lattices with an involutive and order-reversing mapping, as it is the case of those Flexible algebras [15] in which negation is involutive and verifies the duality laws, as well as in its particular case of the Basic Fuzzy algebras [16] ([0, 1] X,, +, ) where these laws hold. Theorem 3.2. In Boolean algebras, =. Page 8 of 12

9 Proof. From the chain, a b a b a + b a b + b a + b a + b a + b a (a + b) a (a + b) a a b a = a b a b, it follows. Provided it is a b, from a b b b = 0 a + b = 1, since b a implies b + b = 1 a + b, and a + b = 1, it follows, and finally, =. Notice that since the first part of this proof just follows from typically Boolean laws, it is neither exactly reproducible in Ortho-lattices, nor in De Morgan algebras. Its validity is reduced to Boolean algebras since, for instance, 1. In the De Morgan algebra ([0, 1], min, max, 1 id), it is = min(0.5, 0.3) = 0.5, but = max(0.5, 0.4) = 0, 5, with 0.5 > 0.4. Of course, it also happens in the extension of such algebra to [0, 1] X, that is, with the only De Morgan algebra of fuzzy sets, and with all the standard algebras of fuzzy sets [11]. 2. In the orthomodular lattice of the vector subspaces of R 3 (with axis x, y, z) endowed with the operations intersection ( ), direct-sum (+), and orthogonal complement ( ) of subspaces, where the order is the inclusion of subspaces, the planes a (given by x and y), and b (given by y and z), verify a b = 0, but a + b = b. Hence it is a b, but not a b. Hence, in De Morgan algebras, in all the standard algebras of fuzzy sets, and in Ortho-lattices actually, and in general, it is not. Thus, and not without a surprise, in Boolean algebras Lewis relation is identical to Russell s relation, there is no difference between them, and both are given by the operations m2 and s2. Indeed, in Boolean algebras there is no way of distinguishing between Russell s and Lewis implication relations. 4 Last remarks a) The only Boolean operation representing the material implication is the usually taken in classical logic, a m2 b = a + b (not a or b), and for the strict implication operation there are the two possibilities a s1 b = a b (a and b, conjunctive implication), and a s2 b = a b + a b (a and b or not a and not b). For what concerns m1, it has the unreasonable property (x m1 b) = (x m1 b), for any x. b) All of the considered implications allow forwards inference, but both strict implications show the property, unreasonable for representing a conditional statement, of being commutative. Page 9 of 12

10 c) It is, (a) a s1 b = 1 a = b = 1, (b) a s2 b = 1 a b. Hence, only the assertion of a s2 b, gives the relation =. d) It is easy to prove that, in Boolean algebras and only in them among Ortho-lattices and De Morgan algebras, it holds a (a b) b a b a + b Thus, = 0, implies (1 0) = 0, for any binary operation satisfying last inequality. Since both material implications mi, and both strict implications si, verify the Modus Ponens inequality, the condition 1 0 = 0 supposed at the beginning is actually superfluous. Even more, it is a + b = 0 a = 1 and b = 0, but this is neither the case with a b, nor with a b + a b. That is, (a) a m1 b = 0 b = 0, (b) a m2 b = 0 a = 1 and b = 0, (c) a = 1 and b = 0 a s1 b = 0, but not reciprocally, (d) a = 1 and b = 0 a s2 b = 0, but not reciprocally, in which case what is obtained is a b = 0 and a + b = 1, that is, the set {a, b} is a partition of B. e) For the four implications in section 2.5, Modus Ponens Inequality holds thanks to the verification of a (a b) = a b. Hence, the reasoning in sections 2.3 and 2.4 could be shortened by taking into account a (α a b + β a b + γ a b + δ a b ) = a b, and 1 1 = 1, with which it immediately follows α = 1, = 0, and rests a b = a b + β a b+δ a b with β, δ {0, 1}. Hence, the four implications also follow by just replacing β = δ = 0, β = δ = 1, β = 0; &; δ = 1, and β = 1; &; δ = 0. Page 10 of 12

11 5 Conclusion Given an operation I in a Boolean algebra B, let s denote by R I the relation defined by a I b = 1. It is R m1 = {(a, 1); a B}, R s1 = {(1, 1)}, R m2 = R s2 = ; thus, R s1 R m1 =, without coincidence between the first three relations. Hence, neither s1, nor m1, define Lewis relation. If relation R s1 is more restrictive than, the same paradoxes than those of material implication can really appear with. If the qualification strict is clearly justified with both operations s1 and s2, only R s1 and R m1 are actually relations stricter than. This paper s only goal is to offer a view on Lewis strict implication in the framework of a Boolean algebra B, by identifying p is necessary with non p is self-contradictory as, in fact, Lewis did. A view that, showing that Lewis relation of implication actually disappears in Boolean algebras, and is given by two implication operations, leaves actually open the question of what happens in less restrictive algebraic structures like, for instance, Ortho-lattices and De Morgan algebras where the only that right now can be concluded is, showing that Russell s is stricter than Lewis relation. Nevertheless, it should be pointed out that both in Ortho-lattices and De Morgan algebras, the validity of the inequality a (a +b) b, forces them to be Boolean algebras [14]. Hence, the problem of finding an operation satisfying the Modus Ponens inequality and giving R =, seems to remain open out of Boolean algebras. Acknowledgments This paper is partially supported by the Foundation for the Advancement of Soft Computing, and by the project MICINN/TIN V References 1. M. Adams: Did Ockham Know of Material and Strict Implication? Franciscan Studies, vol. 33, 1973, Ph. Boehner: Does Ockham Know of Material Implication? Franciscan Studies, vol. 11, 1951, Félix Bou: Complexity of Strict Implication, Advances in Modal Logic 2004: G. Frege: Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens, Halle a. S., 1879 (English: Concept Notation, the Formal Language of the Pure Thought like that of Arithmetics). 5. E. V. Huntington: The Relation between Lewis s Strict Implication and Boolean algebras, Bulletin of the American Mathematical Society, vol. 40, 1934: D. A. Kappos: Probability Algebras and Stochastic Spaces, New York: Academic Press, C. I. Lewis: A Survey of Symbolic Logic. Berkeley 1918, Reprint, New York C. I. Lewis, C.H. Langford: Symbolic Logic (Second edition), New York: Dover, Page 11 of 12

12 9. M. Mullick: Does Ockham Accept of Material Implication? Notre Dame Journal of Formal Logic, vol. 12, 1971: von Megara 11. A. Pradera, E. Trillas, E. Renedo: An Overview on the Construction of Fuzzy Set Theories, New Mathematics and Natural Computation, I(3), 2005, S. Rudeanu: Boolean Functions and Equations, North-Holland: Elsevier, E. Trillas, S. Cubillo: Modus Ponens on Boolean Algebras Revisited, Mathware and Soft Computing, vol. 3(1-2), 1996: E. Trillas, C. Alsina, E. Renedo: On three laws typical of Booleanity, Proceedings NAFIPS, vol 2, 2004: E. Trillas, I. García-Honrado: A Reflection on Fuzzy Conditionals, Combining Experimentation and Theory. A Homage to Abe Mamdani (Eds. E. Trillas et altri), Springer, Berlin, E. Trillas: A Model for Crisp Reasoning with Fuzzy Sets, International Journal of Intelligent Systems, vol.27, 2012, A. N. Whitehead, B. Russell: Principia Mathematica (Second Edition), Cambridge: Cambridge University Press, Page 12 of 12