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| title | date | draft | tags | math |
|---|---|---|---|---|
| Intensional Logic Extends First Order | 2022-02-26T20:33:38-05:00 | false | true |
The second brightest object in the sky is known as the morgensteorra (morning star) and æfensteorra (evening star). Later on this object became known as Venus. (Wikipedia) $$ \text{morgensteorra} = \text{æfensteorra} = \text{venus} $$ Gottlob Frege asks in 1892 whether we should make a distinction between a sense and a reference. (SEP) (Wikipedia)
One might be tempted to think that traditional first order logic can handle this. To show how we'll need to extend it, let us think of this problem from a cognitive perspective. Lets say that we have a relation B that stands for belief. Now lets say that an agent has a belief that Venus is the evening star.
$$
B(\text{æfensteorra} = \text{venus})
$$
In first order logic, we can then deduce the following:
$$
B(\text{morgensteorra} = \text{venus})
$$
But does that make sense? It is possible to hold a belief that Venus is the evening star while not holding a belief that Venus is the morning star. Therefore, we cannot treat belief as a traditional relation symbol. Issues like these give birth to intensional reasoning and from that modal logic.