[MUSIC] Welcome at this lecture, in this lecture, I will speak about dualities and logic. If we apply negation to true we obtain false, and if we apply negation to false, we obtain true. And what we see is that true and false are the same OPX, except that you have to apply a negation on the one to obtain the other. And in such a case we speak about joule constants or joule joules, so false and true are dual to each other. And you can also see that if you look at the and and the or, so if you apply negation to the and, you obtain or. If you apply negation to the or, you obtain the and, and also here we say that and and or are dual to each other. Also for the model operators, we have this nice duality. Namely, if we apply negation to the diamond modality, then we obtain the box modality. And if we apply negation to the box modality, we obtain the diamond modality, and also here we have an example of duals. And logic is full of duels, for instance the universal quantifier is dual to the existential quantifier. And in the next lecture, I will speak about a minimal fixed point operator and a maximal fixed point operator, and they'll also turn out to be dual to each other. So what is one of the advantages of dualities? That is that the properties of one operator is also carrying over to the dual operator. So an operator and its dual often have the same properties, and this can very clearly be seen in the equation. So if we have equations for the and, like commutativity, then we see that this also holds for the or. And associativity for the and carries over to the or also, and item potency for the and is also valid for the or, in the same way as for true. For the and or we can see the discourage offer, to where did you let the false and truth. So the not true is false and you will leave the not false is equal to true. And if you have pi and true is equal to pi then you will leave define pi or false is equal to pi, and this goes on, and on, and you can see that we have all kinds of valid dual formulas. The only identity for which we do not have a dual is that identity nought nought phi is phi because nought does not have its dual. So let's look at the duals in Model fits model properties. So we knew that box a true is equal to true. And we now probably need to recognize that diamond a false is equal to false, is the dual property of the first one. And in the last lecture, we saw that the box a distribute over the ends. And if you recognize now that residual to the property that diamond a distributes over the or. So one of the properties of duals is that, they are expressible using negations or false is expressible, it's not true. The same thing as not true is expressible, it's not false, or as expressible in terms of the ent or. And to expressionable in or if you would like to do that, and you can also see that the book's modality is expressionable. And the diamond modality, or refers to the diamond modality, is expressionable in the book's modality. And this allows us to remove operators from our syntax. So this is our full [INAUDIBLE] syntax, false or superfluous, or superfluous. Books diamond can be removed without changing the expressibility of this language. So we can also replace this full syntax by the smaller syntax, let out, loosing the capacity to express properties. And what we'll see is that for theoretical considerations, it's very useful to have the small syntax. But if he wants to use this language to study systems, then we'd rather prefer the external syntax, so we can express properties in the way we would like to without encoding it into a set of core operators. Okay, let's look at an exercise. Here we have two pairs of expressions. On each line we have two expressions and the question is, are these pairs of expressions dual to each other? So we introduce this notion of dual operators and it's really useful if you are aware of their existence. We showed that lots of equations have that dual part, so we only have to recall 50% of all the equations and the properties of all the operators. Because the properties of an operator are generally the same for each dual. And we saw that we can use dualities to make the syntax very small. And this is particularly useful if we want to theoretically study our language. In the next lecture, I will speak about the all important minimal and maximal fixed point operators, thank you for watching. [MUSIC]