What are the necessary logical conditions for something to be eternal (i.e. without beginning, always existing)?
I can only see three conditions analytically. I'd say one of these is 'necessary', but really what it is doing is elaborating on what is meant by eternal.
It is outside of time so that the notion of time doesn't apply to it.
It is wholly within time. It begins with the beginning of time and it ends with the ending of time.
It overlaps time. It 'begins' before the beginning of time and it 'ends' after the ending of time.
An example of condition 1: In Islam (I don't know enough about theology to say which school exactly) and also the Christian philosopher and theologian Aquinas, say that Allah/God being eternal means He is outside of time. Also, mathematical concepts, if one accepts that they belong to a Platonic realm also lie outside time - as do Plato's theory of forms.
Of 2: There is one thing that is eternal by definition and that is time itself. A second thing that is eternal is the universe itself. Now, even if time were to last 20 billion years only, what could it mean for something to last longer, for there is no more time for it to last longer. This is how Stephen Hawking for example argues that there is nothing beyond the beginning or end of our time.
Of 3: But are we limited by our imagination? Can there not be more than one kind of time? There is nothing to logically insist that there must be only one. All we have is the evidence of our eyes and our inner sense of time. It is in this sense that Spinoza will say that nature is one of the modes of God, that the character of eternity in nature derives from God, but God is the only true eternal substance.
It must be indestructible, but that doesn't mean that it can't undergo change or be composite. (Even an ordinary cat which lives and breathes is permanent and unchanging in some way whilst it obviously changes).
To be honest, all of these ideas are rehearsed and explicated in the notion of substance which was originally articulated by Aristotle for his metaphysics, picked up then by Islamic and Christian theologians and then by modern rationalist science and philosophy. One can consider it to have roots before that. For example, the idea of apeiron (the boundless) by the Milesian philosopher Anaximander who based his cosmology on it.
Assuming: By 'eternal' you mean: "Eternity (or forever) is endless time. In philosophy and mathematics, an infinite duration is also called sempiternity, or everlasting."
Basis: Existence is based on being directly(e.g. senses) or indirectly(e.g. logic) perceived by a self.
Elaboration of the basis: For existence to be confirmed direct or indirect perception is a must.
- Direct perception includes all the sensory inputs irrespective of them being augmented or bare.
- Indirect perception includes all notions of logic, including faith based on logical interpretation of experiences.
- Self includes, apart from others, oneself, the consciousness that validates one's own existence. Oneself after resting in the confirmation of one's own existence, seeks the existence of others.
- Without such perception by a self, existence cannot be confirmed in any way.
My Hypothesis: The only condition for something to be eternal would be:
- To co-exist in perfect harmony with non-existence
Elaboration of the hypothesis: An eternal would not be dependent on the need to be validated/confirmed of its existence by any self. However, it does practice absolute control on confirmation of existence of itself, and hence the others. It therefore cannot be categorized as a sentient or a non-sentient existence. It is either, both and none, as per its own free will, in following ways:
Existent: It confirms its own existence and seeks others which includes responsiveness to stimuli
Non-existent: It negates its own self and hence if perceived by others appears to be non-existent
Both: It negates its own self but confirms the existence of others by including them in its-self that is been negated OR It negates the existence of others but confirms its own existence by seeking the confirmation of its-self alone, incessantly.
None: It does not perceive any self at all
Time is perceived indirectly by the self. Hence, something that can outlast the self is eternal.