website/content/blog/intensional-logic-extends-first-order.md

date	draft	math	medium_enabled	medium_post_id	tags	title
2022-02-26 20:33:38-05:00	false	true	true	8e07b68aa8f5	Logic	Intensional Logic Extends First Order

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.