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Radhakrishnan Srinivasan - One of the best experts on this subject based on the ideXlab platform.

  • LOGICAL ANALYSIS of THE BOHR COMPLEMENTARITY PRINCIPLE IN AFSHAR'S EXPERIMENT UNDER THE NAFL INTERPRETATION
    International Journal of Quantum Information, 2010
    Co-Authors: Radhakrishnan Srinivasan
    Abstract:

    The NAFL (non-Aristotelian finitary logic) interpretation of quantum mechanics requires that no "physical" reality can be ascribed to the wave nature of the photon. The NAFL theory QM, formalizing quantum mechanics, treats the superposed state (S) of a single photon taking two or more different paths at the same time as a logical contradiction that is formally unprovable in QM. Nevertheless, in a nonclassical NAFL model for QM in which the Law of Noncontradiction fails, S has a meaningful metamathematical interpretation that the classical path information for the photon is not available. It is argued that the existence of an interference pattern does not logically amount to a proof of the self-interference of a single photon. This fact, when coupled with the temporal nature of NAFL truth, implies the logical validity of the retroactive assertion of the path information (and the logical superfluousness of the grid) in Afshar's experiment. The Bohr complementarity principle, when properly interpreted with t...

  • LOGICAL ANALYSIS of THE BOHR COMPLEMENTARITY PRINCIPLE IN AFSHAR'S EXPERIMENT UNDER THE NAFL INTERPRETATION
    International Journal of Quantum Information, 2010
    Co-Authors: Radhakrishnan Srinivasan
    Abstract:

    The NAFL (non-Aristotelian finitary logic) interpretation of quantum mechanics requires that no "physical" reality can be ascribed to the wave nature of the photon. The NAFL theory QM, formalizing quantum mechanics, treats the superposed state (S) of a single photon taking two or more different paths at the same time as a logical contradiction that is formally unprovable in QM. Nevertheless, in a nonclassical NAFL model for QM in which the Law of Noncontradiction fails, S has a meaningful metamathematical interpretation that the classical path information for the photon is not available. It is argued that the existence of an interference pattern does not logically amount to a proof of the self-interference of a single photon. This fact, when coupled with the temporal nature of NAFL truth, implies the logical validity of the retroactive assertion of the path information (and the logical superfluousness of the grid) in Afshar's experiment. The Bohr complementarity principle, when properly interpreted with the time dependence of logical truth taken into account, holds in Afshar's experiment. NAFL supports, but not demands, a metalogical reality for the particle nature of the photon even when the semantics of QM requires the state S.

Inoue Kazumi - One of the best experts on this subject based on the ideXlab platform.

  • Dialectical Contradictions and Classical Formal Logic
    International Studies in the Philosophy of Science, 2014
    Co-Authors: Inoue Kazumi
    Abstract:

    A dialectical contradiction can be appropriately described within the framework of classical formal logic. It is in harmony with the Law of Noncontradiction. According to our definition, two theories make up a dialectical contradiction if each of them is consistent and their union is inconsistent. It can happen that each of these two theories has an intended model. A number of examples of this are to be found in the history of science.

Edward N. Zalta - One of the best experts on this subject based on the ideXlab platform.

  • In Defense of the Law of Noncontradiction
    2006
    Co-Authors: Edward N. Zalta
    Abstract:

    An important philosophical puzzle arises whenever we find a group of philosophically interesting sentences which individually appear to be true but jointly imply a contradiction. It is traditional to suppose that since the sentences in the group are jointly inconsistent, we cannot accept them all. This refusal to accept all the sentences in the group is not just grounded in (a) the problem of accepting the derivable contradiction, but also in (b) the problem that classical logic gives us the means to derive every sentence whatsoever once we have derived a contradiction. But with certain really hard puzzles of this kind, it is difficult to identify even one sentence in the puzzling group to reject. In such cases, there seems to be no good reason or argument for rejecting one of the sentences rather than another. We often find ourselves in the uncomfortable position of having to reject statements that have a strong claim to truth. Paraconsistent logic and dialetheism constitute a fascinating body of doctrines for critically analyzing this kind of philosophical puzzle. Paraconsistent logic removes problem (b), noted above, concerning the presence of contradictions in classical logic. In contrast to classical logic, paraconsistent logic tolerates the derivation of a contradiction without thereby yielding a proof of every sentence. Dialetheism goes one step further, however, and addresses problem (a). It is the doctrine that, in some of these really hard cases, there are indeed true contradictions. Di-

Brian Huss - One of the best experts on this subject based on the ideXlab platform.

  • Cultural differences and the Law of Noncontradiction: Some criteria for further research
    Philosophical Psychology, 2004
    Co-Authors: Brian Huss
    Abstract:

    Recent psychological research on the connection between culture and thought could have dire consequences for the idea that there are objective standards of reasoning and that meaningful cross-cultural discussion is possible. The problems are particularly acute if research shows that the Law of Noncontradiction (LNC) is not a universal of folk epistemology. It is extremely difficult to provide a non-circular justification for the LNC, and yet the LNC seems to act as a basic standard for reasoning in the West. If non-Western cultures do not believe the LNC holds, then meaningful cross-cultural discussion and debate will be very difficult, to say the least. In this paper it is argued that the distinction between belief and acceptance is important in analyzing cross-cultural studies on the way people reason. Studies conducted by Richard Nisbett and Kaiping Peng concerning differences between East Asians and Westerners are analyzed. The distinction between belief and acceptance is used to demonstrate that the ...

Jean-yves Béziau - One of the best experts on this subject based on the ideXlab platform.

  • Idempotent Full Paraconsistent Negations are not Algebraizable
    Notre Dame Journal of Formal Logic, 1998
    Co-Authors: Jean-yves Béziau
    Abstract:

    Using methods of abstract logic and the theory of valuation, we prove that there is no paraconsistent negation obeying the Law of double negation and such that ¬(a ∧ ¬a) is a theorem which can be algebraized by a technique similar to the Tarski-Lindenbaum technique. 1 What are the features of a paraconsistent negation? Since paraconsistent logic was launched by da Costa in his seminal paper [4], one of the fundamental problems has been to determine what exactly are the theoretical or metatheoretical properties of classical negation that can have a unary operator not obeying the principle of Noncontradiction, that is, a paraconsistent operator. What the result presented here shows is that some of these properties are not compatible with each other, so that in constructing a paraconsistent negation as close as possible to classical negation, we have to make a choice among classical properties compatible with the idea of paraconsistency. In particular, there is no paraconsistent negation more classical than all the others. The incompatibility appearing here is between theoretical properties (double negation and ¬(a ∧ ¬a) as a theorem) and a metatheoretical property (replacement theorem). One who chooses the theoretical properties will not be able to algebraize his system with the usual Tarski-Lindenbaum method and should use some alternative treatments such as that in da Costa [5]. On the other hand, one who chooses the metatheoretical property will have to sacrifice at least one fundamental theoretical property of negation, risking the possibility of dealing with an operator that is a modality rather than a negation. The result presented here is of the same kind as some previous results concerning the incompatibility between the replacement theorem and the paraconsistent logic C1 of [4]. It was soon realized that the replacement theorem is not valid in C1. Mortensen [14] proved that, in fact, it was impossible to define a nontrivial congruence in C1. Urbas [19] proved that the addition of the replacement theorem to Received March 18, 1997; revised June 1, 1998 136 JEAN-YVES BEZIAU C1 leads to classical logic or trivializes it. (However, Mortensen [15] and da Costa, Beziau, and Bueno [8] have presented extensions of C1 which admit nontrivial congruences.) The question whether the replacement theorem is compatible or not with the idea of paraconsistency is one of the significant remaining problems in paraconsistent logic (Beziau [2]). 2 Basic framework Generalizing the definition of Suszko (cf. [3]), we consider an abstract logic (or simply, a logic) as a structure L = 〈G;Cn〉 where G is a structure of domain S and Cn a closure operator on S (i.e., for every subset A and B of S, A ⊆ Cn A, CnCn A ⊆ Cn A, A ⊆ B =⇒ Cn A ⊆ Cn B). Thus, G is not necessarily an absolute free algebra as in standard propositional logics: it can, for example, be a partial infinitary algebra or a relational structure. The result that we will prove here can be applied to firstor second-order language and even to non-well-founded language as described by Lismont in [12], because it does not depend on specific features of the structure G, which is the mathematical expression of the language. Given a family F of subsets of a set A, it is easy to see that the function φ on P (A) × P (A) defined by: for every a ∈ A, A ⊆ A, a ∈ φA if and only if for every B ∈ F, A ⊆ B =⇒ a ∈ B is a closure operator on A. Such a family will be called an adequate bivalent semantics (ABS) for an abstract logic L = 〈G;Cn〉 when the domain S of G is the same as A and φ is the same as Cn. An element of the family is called a bivaluation because we consider its characteristic function. It is easy to prove that the class of ABS for a given logic is not empty. This justifies the study of a logic from the point of view of its class of ABS, called the theory of valuation and developed by da Costa (for an overview on the subject see [6] and [7]). 3 Preliminary results A negation in an abstract logic L = 〈G;Cn〉 is a unary operator ¬ on the domain S of G with certain properties. Until now there has been no agreement concerning these properties, but it is common, for example, to call a negation an operator not obeying the Law of excluded middle, such as that involved in intuitionistic logic. Thus, it is not necessarily absurd to call a negation an operator not obeying the Law of Noncontradiction. A negation ¬ is paraconsistent if and only if there exists a in S such that Cn{a,¬a} = S (a paraconsistent logic is a logic with such a negation). We will state an easy result establishing a connection between this definition and the intuitive traditional formulation of the Law of Noncontradiction (a proposition and its negation cannot both be true) and which will also be useful for the proof of the central theorem. Before doing so we need two more definitions. A bivaluation β in an ABS for the logic L is singular if and only if there exists an object a in S such that β(a) = β(¬a) = 1. A singular bivaluation is trivial if and only if it is the function which gives the value 1 to any object of S. Proposition 3.1 A logic is paraconsistent if and only if in all its ABS there is a singular nontrivial bivaluation. Given a logic L = 〈G;Cn〉, two objects a and b of S are logically equivalent PARACONSISTENT NEGATIONS 137 (notation: a ∼= b) if and only if Cna = Cnb. Let us say that an object a is a theorem if and only if Cna = Cn∅. Obviously we have the following proposition. Proposition 3.2 If a and b are theorems, then a ∼= b. Moreover, given S an ABS for L , we have this proposition. Proposition 3.3 a ∼= b if and only if for any β of S, β(a) = β(b). An idempotent negation ¬ of L is a negation such that for every object, a ∼= ¬¬a. An algebraizable negation is a negation ¬ verifying: for every a and b, if a ∼= b then ¬a ∼= ¬b. An algebraizable logic is a logic such that logical equivalence is a congruence on it. It is clear that a logic in which there is a nonalgebraizable negation is not algebraizable. In the case where the structureG is an absolute free algebra or something similar, such as a first-order language, it is easy to see that a logic is algebraizable in the above sense if and only if the theorem of replacement (as formulated, for example, by Kleene in [11]) holds. In the general case, this notion of algebraization is equivalent to the general replacement theorem formulated by Curry and MacLane (see [9] and [13]). Proposition 3.4 Given an idempotent algebraizable negation, two objects are logically equivalent if and only if their negations are logically equivalent. 4 Main result A full paraconsistent negation is a paraconsistent negation such that ¬(a ∧ ¬a) is a theorem. Here we take ∧ to denote the standard conjunction, that is, Cn{A, a, b} = Cn{A, a ∧ b}. It is easy to check then that for any bivaluation β of a given ABS for the underlying logic, we have β(a) = 1 and β(b) = 1 if and only if β(a ∧ b) = 1. Theorem 4.1 Idempotent full paraconsistent negations are not algebraizable. Proof: Let L be a logic with a full idempotent paraconsistent negation. Given S an ABS for L , due to Proposition 3.1, there is a singular nontrivial bivaluation β of S, such that β(a) = β(¬a) = 1 for one object a, and we have β(a ∧ ¬a) = 1. As ¬ is a full paraconsistent negation, ¬(a ∧ ¬a) is a theorem, and given any object b, ¬(b ∧¬b) is also a theorem; therefore, due to Proposition 3.2, ¬(a ∧¬a) ∼= ¬(b ∧ ¬b). Now, due to Proposition 3.4, a ∧ ¬a ∼= b ∧ ¬b, therefore, due to Proposition 3.3, β(b ∧ ¬b) = β(a ∧ ¬a) = 1, thus β(b) = β(¬b) = 1. This shows that β is trivial, which is absurd. 5 Applications of the theorem The paraconsistent negations of Asenjo’s calculus of antinomies (cf. [1]), of D’Ottaviano and da Costa’s logic J3 (cf. [10]), and of Priest’s logic LP (cf. [16]) are defined with Łukasiewicz’s L3 table for negation taking 2 and 1 as designated. It is easy to see that, together with the standard conjunction (included in these logics), this defines idempotent full paraconsistent negations. Therefore these negations are not algebraizable. Moreover, due to our result, it is easy to see that there are no extensions of Asenjo’s calculus, J3 and LP, in which their negations are algebraizable and still paraconsistent. 138 JEAN-YVES BEZIAU 6 Generalization of the theorem This theorem depends on the definition of an abstract logic L as a structure 〈G;Cn〉 which is quite general; however, alternative, more general definitions can be proposed such as a structure 〈G; Xn〉 where Xn is a relation on P (S) × P (X) obeying axioms extending those of Cn as studied by Scott in [18]. Our result can be extended without difficulty to such a structure. We can also generalize this result to the case where we deal not with a closure operator but an equivalence connective, in order to satisfy relevantists who think that the closure operator is wrong (cf. Routley in [17]). Acknowledgments This work was carried out when the author was supported by a grant from the LNCC/CNPq. The author would like to thank M. V. Kritz for his kind invitation from the LNCC as well as an anonymous referee for useful comments on an earlier version of this paper.