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Possible worlds of belief

What are possible worlds of belief?

Let's say that we have a fact X. We may know that X is true. But we might also now that X will be true at a time t in the future. Also X may be true at a time in the future. X is believed to be true.

All this statements can be implemented in logic using possible worlds approach. In a certain world John lives in America. In another world John lives in Africa. Facts can have different truth values, but all worlds may connect to each other.

If fact X is true in all worlds, then we can conclude that X is true.

Such a statement is called a tautology.

In Kripke semantics for possible worlds belief, every world has possible accessible worlds. For example if John lives in America in world 1, then if he believes that is possible that he lives in Africa, then world 2 is accessible from his world, world 1. We can write this as world 1 -> world 2.

Further we can discuss links between more worlds and problems of accessing a world starting from a not directly accessible world. In such cases, there are chains of beliefs that can be modeled.

Properties of the accessibility relation:

1. Reflexivity: w -> w . This means any world is accessible from itself.

2. Transitivity: If there is w1->w2->w3 then we can assume we have w1->w3.

One interesting consequence of the properties, if the model accepts them, is introspection. If John believes X, then John believes that "John believes X" . This is part of Doxastic logic.

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