Principle of Non-Arbitrariness

given: The Law of Non-Contradiction states that a proposition cannot be both true and false at the same time.
The Law of the Excluded Middle states that a proposition must be either true or false.

1. A proposition is either arbitrary (being without rational basis as sufficient explanation for being true or false) or non-arbitrary (being with rational basis as sufficient explanation for being true or false).

2. If a proposition is arbitrary, it would not be logically necessary in any possible world, as it would be with no rational basis for one being affirmed true or false. Any proposition that is not grounded in a sufficient explanation for being true or false in any possible world, would not be rationally possible in any possible world, and could not be true or false.

C. Therefore, every proposition that is either true or false must be non-arbitrary, as it must be with rational basis as sufficient explanation for being true or false in any possible world. It would not be rationally possible for any proposition to be arbitrary. Every proposition must be non-arbitrary.
This is known as the Principle of Non-Arbitrariness.



Consider the standard form for the Principle of Necessitation:

given: The Law of Non-Contradiction states that a proposition cannot be both true and false at the same time.
The Law of the Excluded Middle states that a proposition must be either true or false.

1. A proposition that is possibly necessary in some possible world is a proposition that is necessarily true in at least one possible world.

2. If a proposition that is possibly necessary in some possible world is not necessarily true in every possible world, it would be logically possible for that proposition to be false in at least one possible world, which would make the proposition incoherent and not necessarily true in any possible world.

C. Therefore, if a proposition is possibly necessary in some possible world, it must be necessary in every possible world.
This is known as the Principle of Necessitation.