Paraconsistent logic and contradictory viewpoints

We start by recalling the definition of contradiction from the perspective of the square of opposition, emphasizing that it comes together with two other notions of oppositions, contrariety and subcontrariety. We then introduce the notion of paraconsistent negation as a non-explosive negation; we explain the connection with subcontrariety and why it is better not to talk of contradiction in case of paraconsistent negation. We then explain that we can interpret the paradoxical duality wave/particle either as a subcontariety in reality or as different contradictory viewpoints. We go on developing a logic based on a relational semantics with bivaluations conceived are viewpoints and in which we can define a paraconsistent negation articulating the oppositions between viewpoints. After proving some basic results about this logic, we show the connection with modalities: we are in fact dealing with a reconstruction of S5 from a paraconsistent perspective and our paraconsistent negation is the classical negation of necessity. We finish by presenting a hexagon of opposition describing the relations between this negation, the negated proposition, classical negation and necessity. 1. Contradiction and the square of opposition According to a general informal definition, two propositions p and q are said to be contradictory iff they cannot be true together and they cannot be false together. In this case we say that p contradicts q and that q contradicts p, and also that p and q together is a contradiction. Within the theory of the square of opposition, contradiction is an opposition among other ones: contrariety and subcontrariety. Two propositions are said to be contrary iff they can be false together but not true together. Two propositions are said to be subcontrary iff they can be true together but not false together. Contradiction is considered as the strongest opposition and the relations between the three oppositions can be nicely expressed by a diagram : In this diagram we have represented contradiction in red, contrariety in blue and subcontrariety in green. The black arrows are implications, traditionally called subalternations, and the four letters A, E, I, O, which also are traditional notations, can be considered as four propositions (More about the square of opposition, its history, philosophy and technical aspects, can be found in the two recent books: Beziau and Payette, 2012; Beziau and Jacquette 2012) . What kind of example of contradiction can we give? Is the pair «Kelly is sad» and «Kelly is happy» a contradiction? One may argue that these two propositions are just contrary because Kelly may be neither sad nor happy. One may also argue that these two propositions are subcontrary, because Kelly may be sad and happy at the same time, smiling and crying as it sometimes happens. One may also even claim that Kelly can be neither happy nor sad and both happy and sad. We are then in a paranormal situation: «Kelly is sad» and «Kelly is happy» are neither contrary, nor subcontrary, nor contradictory. Such kind of situation is not described in the square and one may wonder if in this case there really is an opposition between the two propositions (about paranormal negation and paralogics see Beziau 2012b). Let us have a look at less controversial cases where no emotions are involved, let us enter the realm of mathematics. We can consider the two following propositions: «K is a circle» and «K is a square». Using the theory of the square of opposition, we see that these two propositions are in fact just contrary because K can be neither a circle, nor a square, a triangle for example. Extracting properties from these two propositions, we can say that a round square is not a contradictory object, but just a contrary object. Let us try to find a better example: «K is odd» and «K is even». An integer can in fact not be both odd and even. And it has to be odd or even: there is no third possibility, you can divide it by two or not, even an eccentric integer like zero does not escape to it. Now if we consider that K is a number which is not an integer, the situation is not clear, maybe we are facing just contrary propositions, like when we are going out of the realm of mathematics: if K is Kelly, she is neither odd nor even (unless we are think of dividing her in two). To find “real” contradictory propositions we maybe have to go at a superior level of abstraction, to avoid ambiguity due to vagueness and/or contextualization of concepts. We can consider a proposition and its negation, like «Kelly is happy» and «Kelly is not happy». Is it possible to argue that these two propositions are not contradictory. In fact everything is possible, up to the limits of absurdity and triviality. This is the way to paraconsistent logic. 2. Contradiction and paraconsistent negation The idea of parconsistent logic is to reject the principle according to which from a proposition and its negation we can deduce any proposition, a principle traditionally called ex-falso sequitur quodlibet and in modern times, explosion principle. This principle can be symbolically represented as follows: p, ¬p ╞ q The symbol ¬ represents negation and the symbol ╞ represents semantical consequence. Following the standard Tarskian notion of consequence (a general notion not limited to a special logical system)