Final answer:
Modal logic is intensional because it deals with possibility and necessity and considers different possible worlds, which adds a contextual layer to the meaning of statements. The use of modal operators like 'necessarily' and 'possibly' highlights the intensional nature of propositions in modal logic, as the truth value can vary across different scenarios.
Step-by-step explanation:
Modal logic is considered intensional because it deals with the concepts of possibility and necessity rather than actuality. In classical logic, the truth of a proposition is determined by the actual world, but in modal logic, we also consider propositions in terms of how they relate to other possible worlds. This leads to a focus on the intentions or the mode of truth behind assertions, hence the term 'intensional,' to reflect that meaning may vary depending on various possible circumstances or worlds.
In examining modal logic, we find that it includes additional operators, such as 'necessarily' and 'possibly,' which are not concerned with the specific content loaded but rather with the mode of truth of the propositions these operators are applied to. These modal operators demonstrate that truth can be non-extensional, meaning that the truth value of a proposition can vary across different possible worlds.
For example, the statement 'It is possible that it will rain tomorrow' has a different intensional context compared to 'It is necessary that 2+2=4,' because the former is contingent upon circumstances in the actual world, while the latter is a necessary truth in all possible worlds. The distinction between necessary and contingent truths, or between what could be true and what must be true, is a hallmark of what makes modal logic intensional.