I saw this question here https://latin.stackexchange.com/questions/18991/translating-all-things-come-to-an-end-to-latin and immediately I thought "if all things come to an end, then the action of all things ending eventually itself comes to an end".
Is this a logical paradox?
The problem with this question is that all things do not come to an end.
For example: integers, rational, irrationals - the lists of each never ends.
If you recast the question as: All things that are not infinite come to an end, then, when they have all finished, 'all things coming to an end' will also, naturally, finish.
Unless, the things in question are able to create new instances of things, in which case the answer is: no.
I think of "all things coming to an end" as a singular moment. A snap of the fingers. In this case, the problem is easily bypassed as "all things coming to an end" is, in and of itself, the end.
[I]f all things come to an end, then the action of all things ending eventually itself comes to an end.
If this is a paradox, it is a mild one which can be resolved by understanding that it's easy to conflate your two uses of the word 'end'. You have two propositions joined by the material conditional. Let's pull them apart.
- All things come to an end.
- The action of all things ending eventually comes to an end.
Let's try interpreting it your sentence with something concrete.
If a speech comes to an end, then the ending of the speech eventually comes to an end.
Ah ha, this sounds like it might be paradoxical. How can the end of something itself have an end, right? Except, it's not if you understand you are equivocating on 'end'. Really, there are two different senses. Let's rewrite:
If a speech comes to an end-of-speech, then the ending of the speech eventually comes to an end-of-ending.
There you have it! The sense of end-of-speech clearly is different than the sense of end-of-ending. How this works semantically is that end-of-speech is conceptualized as a point in one context, the context of first proposition, and a range in the second proposition that lasts from beginning-of-end to end-of-end. We do this all the time when talking about events (SEP):
The intuition that events are properties of times can also be fleshed out in terms of thinner metaphysical commitments, by construing events simply as times cum description, i.e., as temporal instants or intervals [emphasis mine] during which certain statements hold (van Benthem 1983).
So, in proposition 1, the semantics imply that the end of a speech is a temporal instant, but in 2, the semantics imply that the end of a speech is an interval. In other words, we are mapping an interval onto a point, something that is mathematically quite savvy. Consider tensor fields, for instance, which map tensors onto points. Same exact idea. You might ask, but if Euclid defined a point as having no dimension, how can it be equivalent to a tensor which is analogous to a space? Isn't that a contradiction? Well, yes and no.
If the end of a speech is an instant and an interval, then there is a contradiction of sorts. How can it be two things at once? Well, think of it this way, a point can function like a variable name bound to a variable domain of discourse. Thus a variable name (let's say K) can be viewed as a constant at any moment (K1:=1), but also understood as a representing all naturals (Kn:=n | n is a member of an index set of naturals and the naturals themselves). That's the beauty of sense and reference. We can have a name K, but we can have both different Ks and different references to which K refers. We do it all the time in math, computer science, and logic, to conflate something particular with something universal and that's okay, because whether it is particular or universal in meaning is determined by context.
In your sentence, having a temporal instant lets us talk about the sequence (before-event, event, after-event) in a tidy manner. The event has an end and after-event has a beginning that (roughly or exactly) correspond to the same instant. But, using fuzzy logic, we can acknowledge that picking an exact instant in time is messy, so we can also choose to conceptualize an interval. Again, this is something that we do famously in calculus when we add rigor to the definition of a limit using the epsilon-delta definition (calcworkshop.com).
So, is there a paradox? Yes, but a small one that is easily (and routinely) dissolved. Remember, a paradox isn't a logical contradiction per se because it is MORE than a contradiction. It is the appearance of meaningless when failing to take into account adequate context, and the psychological confusion and cognitive dissonance that results. In fact, logical contradictions even as dialetheia can be quite meaningful.
If the set of all things is finite, and new things are not created forever, then yes. Once the last thing ends and nothing else is ever created, things ending ends.
If you see “all things end” as a thing itself, and the statement is true, then “all things end” is the very last thing that ends.
