On the Smarandache Paradox Abstract

Leonardo Motta [email protected] Conselheiro Furtado, J 574/50 I Helem, PA 66040-100, Hra=il The Smarandache Paradox is a very interesting paradox of logic because it has a background common sense. However, at the same time, it gets in a contradiction with itself. Although it may appear well cohesive, a careful look on the science definition and some logic can break down this paradox showing that it exist only when we are trying to mix two different universes, where in one we have two possibilities and in the other we have only one. When we try to understand the second possibility in the universe which has only one possibility, we end in the Smarandache Paradox. 1. On the Smarandache Paradox The Smarandache paradox can be enunciated as follows. Let A be some attribute (e.g., possible, present, perfect, etc.). If everything is A, than the opposite of A must be also A! For example, "All is possible, the impossible too" and "Nothing is perfect, not even the perfect" [1]. This paradox is very interesting because it has its logic but it makes no sense at the same time. It's very easy to break down this paradox by simply taking a careful look on the definition of science. Ifwe have two possible states, but the whole universe is immerse on only one of the states, then there's no sense talking about another state. It does not exist at all. The same logic can be applied on the Smarandache paradox. Let P be a Boolean property (i.e., it assume only "true" or "false", "0" or "1 ", etc., as value). Now, suppose we have a group of particles that have this property, where half particles are P = true and the other half P = false. Restrictively in this group, say G l' its possible to have both properties. However, if we define another group, G2, where all particles are P = true, then it makes no sense talking about P = false in that group because that property does not exist at all. In other words, if all is possible, then it makes no sense saying that even the impossible is, because in this group we do not have any impossibility. It's a mistake trying to say that even the P = false is P = true in G2 because in that universe P does not assume any value different from "true". Moreover, if we have a group where nothing is perfect, then the perfection does not exist in that group. The paradox simply tries to get the perfection, which is possible in other groups, to this particular group where it is not. Thus, a truth for a particular group may be not true for another one.