An interesting topic in metaphysics is the concept of necessity. Without any pretension of being exhaustive, in this post I'll comment on some of the main elements and kinds of necessity.
Kinds of necessity:
1-Logical, formal or conceptual necessity: This kind of necessity refers to truth-values of propositions, and have not (or doesn't need) to have any ontological commitments. They're truth in virtue of the concepts being used in the proposition.
For example, "bachelors are unmarried men" is necessarily true, in virtue of the concept of "bachelor". But note that this proposition doesn't say anything about the objective existence of bachelors. The proposition is true regardless of the existence of any bachelor.
For example: In classical theism, "God is omniscient" is necessarily true, in virtue of the concept of God. Again, this proposition tells us nothing about the objective existence of God. The proposition is true even if God doesn't exist.
For example: "2+2=4". This mathematical proposition is conceptually true (given Peano's axioms, the rules of inference of standard aritmethics and the concepts of "2", "+" and "4"), even if numbers don't exist objectively.
Note that all of these propositions don't need to have any ontological commitments. Metaphysically, they are all subjective, in the sense of being mind-dependent (they depend for their existence on the concepts used by a mind. No mind --> no concepts --> no proposition --> no truth-values).
Since all the above necessarily true propositions don't imply the objective existence of the entities being mentioned in the proposition, then it follows that if such entities exist objectively, the question of "what explains these entities?" still makes sense.
For example, despite of "bachelors are unmarried men" being (conceptually) necessarily true, it doesn't explain why the hell bachelors exist at all. In fact, the existence of bachelors is (metaphysically, see below) contingent.
2-Metaphysical or ontological necessity: This kind of necessity refers to the existence of objects. Some object X is ontologically necessary if it couldn't not to exist. With some qualifications, they don't have anything to do with the concepts being used in propositions. They don't depend for their existence on the contents of the mind.
Philosophers specialized in metaphysics have realized that necessarily existing beings (e.g. God or numbers, if they exist) must have, in virtue of their necessity, at least the following two features:
-Eternity: they could not have a beginning nor end
-Indestructibility: They cannot be destroyed, otherwise they could have an end and hence could become non-existent. (This is why, Victor Stenger's "metaphysical pizza" reply against the ontological argument is clearly stupid, self-contradictory and embarassing, and only shows Stenger's intellectual limitation to understand propertly philosophical concepts).
Both properties are essential to necessity, not accidental. Lacking one of them implies that the being in question is not necessary, but contingent.
Some implications:
Some important implications follow the above distinction:
1-Conceptually necessary truths don't have (or don't need to have) ontological commitments. This means that if a given conceptual truth is necessarily true, it doesn't mean that the entities posed by the propositions exist objectively.
For example, "triangles have three angles" is necessarily true, even if not triangle exists at all. And the existence of physical triangles in the world could be purely contingent and in need of further explanation (e.g a triangle on the sand requires of the existence of sand, which is a purely contingent phenomenon explained by natural sciences).
2-Some conceptually necessary truths could pose entities which exist in a metaphysically necessary way.
For example "God is omniscient" is conceptually and necessarily true, but it doesn't tell us if God exists or not. But, if God exists, then it exists necessarily. So the proposition is both "conceptually necessary" and poses an entity which exists necessarily.
3-If an object is not eternal, it is not necessary.
In all the history of thought, atheism has hold that the universe is eternal and necessary and hence without a cause.
The problem for atheism is that contemporaty cosmology tells us that the universe began to exist 13.798±0.037 billions of years ago. Therefore, the universe is not eternal. Therefore, the universe is contingent and it needs some kind of explanation of "why does it exist?" like any other contingent being.
This is why some contemporary atheists are prepared to say that the universe began to exist from "nothing", in order to avoid theism.
4-Metaphysical truths don't need to be conceptually necessary:
For example, "whatever begins to exist has a cause" seem to be metaphysically necessary, since its denial implies the coming into being "out of nothing", which seems to be impossible (except for atheists).
However, such proposition is not conceptually necessary, i.e. it doesn't derive its truth-value from the concepts being used in the proposition.
For example, "an actual infinite cannot exist in the concrete world" is not a conceptually necessary truth. But, given the absurd and physically impossible consequences that it has if it were instantiated in the real concrete world, it seems to be metaphysically impossible.
5-Only a necessary being could be the ontological ground for other metaphysically necessary beings.
In other words, necessary entities cannot be ultimately grounded (metaphysically) in contingent beings.
So, if moral values exist necessarily, they cannot be grounded in the physical universe (because it is contingent). This is why atheists tend to hold either the non-existence of such values (the most likely and consistent position given atheism), or to hold that such values exists contingently as an emergent property of matter (something which no natural science supports, since natural science have not discoveried any moral property in physical matter and all the laws of matter have nothing to do with morals nor other personal properties whatsoever).
But a necessarily existing personal being, like God, could ground the existence of necessary moral values (e.g. if God is the "Good", as classical theism holds, moral values must exists eternally and necessarily. God's nature implies them). But not all of morals need to be necessary. God could create them contingently too (e.g. specific moral laws for specific universes, in the same way that physical laws are created for specific universes or realms of existence). Both possibilities are open, provided that God exists and creates universes.
Since moral values are connected to persons, if morals are metaphysically necessary, then persons are metaphysically necessary too. This metaphysically necessity of persons (at least one person) is precisely what theism holds, and what atheism historically has denied and continue to deny (for contemporary metaphysical atheism, persons are not basic to reality, but later by products of a more fundamental non-personal reality, like quantum particles, fields, wave functions, etc. or in the case of some Eastern atheistic worldviews, "impersonal consciousness", "all-pervading energy", Deepak Chopra's "impersonal intelligence" and so forth).
Something similar could be said of mathematical objects (numbers, sets, etc.) but, in contrast with morals, numbers don't seem to be essentially connected with persons, except for the fact that numbers seem to be the contents of some mind.
So, if numbers exist objectively, the following conclusion of mathematician Elliot Benjamin, PhD, seems to be reasonable:
Well if it were the case that numbers and mathematics did exist in some kind of objective/ontological sense, then perhaps this would give us some evidence for some kind of intelligent being who designed the universe--I suppose you can call it God. For the astounding logic involved in higher mathematics is staggering virtually beyond comprehension, with a phenomenal level of mental acrobatics involved in the highest mathematical realms.
This would lead us into a complex mathematical argument for God's existence (there are several of them), which is not the purpose of this post (see, however, this post).
In future posts we'll continue discussing the problems of necessity.
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