In regards to the universe if it said to have a start (Beginning) does that imply that it will have an end? Are beginning and ends implied when either ones is said to be true of a proposition or can we have an eternal universe with an end or a universe with a beginning and no end?
The general view amongst Cosmologists is that the fate of the Universe depends on its overall shape or geometry.
The shape of the universe depends on the density of matter contained therein and is usually expressed using a parameter called the density parameter, which is the ratio of the observed density to the critical density. (Naively, one might say the more matter the more gravitational pull, while a smaller density of matter means there is not enough gravity to pull it all back together.)
The critical density is currently estimated to be approximately 5 atoms (of hydrogen) per cubic meter.
The observed density of ordinary matter is estimated at 0.2 atoms per cubic meters. This leaves us having to estimate the density of the exotic dark energy.
If the ratio is greater than 1 (high density, positive curvature), then the geometry of space is closed, meaning that it has a finite future. This case corresponds to our universe having a spherical shape.
If the ratio is less than 1 (low density, negative curvature), then the geometry of space is open, meaning that it has an infinite future. This case corresponds to a saddle shaped universe.
Finally, if the ratio equals 1 (the "Goldilocks" scenario, no curvature), then the geometry of space is flat, as in the Euclidean geometry we learned in school. Here, the universe expands forever, but the rate of expansion decreases and asymptotically (tends to) zero (over time).
When dark energy estimates are included, our current best guess is that our universe is within 0.4% of being flat, implying an infinite future.
Regarding the past, our science and observation strongly suggest that our universe has a finite past.
In recent years, a multiverse view of reality has become mainstream. Here, our universe is just one instance of many universes that make up a multiverse. In this vision, the multiverse is infinite in both directions.
Current cosmology, largely based on General Relativity, allows the universe could have a beginning in the sense of extending only finitely far into the past, and could have an end in the sense of extending only finitely far into the future, or both, or neither. Currently observation is taken to suggest a finite past going back to a "big bang" though it could also be that the universe vanishes asymptotically over an infinite past. And it is taken to indicate an infinite future, but this could possibly change as new ideas of "dark matter" and "dark energy" emerge. Of course those scientific ideas could all change if currently unknown data come in.
On the other hand if you want to know what Aristotle, or Kant, or Mahayana Buddhism say about it, those are other questions.
I would say that "Does A imply B?" is equivalent to: "Are there zero valid models where A is true but B is not?" If there are valid models where A is true and B is not, then it is not the case that A implies B. This is a useful definition of the word "implies".
Are there valid models where the universe has a beginning but does not have an end? Yes. Therefore it cannot be the case that the universe having a beginning implies it must have an end.
Do we live in a universe that had a beginning but will not have an end? That is a different question. My understanding of the data available is that the answer to this question is "probably".
Well, I don't think there can be such things as "beginning" or "end". There are just sequences of events, we choose to call it "beginning" or "end". For example, suppose I kick a piece of sand in the desert; we may classify my leg going in the direction of the sand as the sequence of events which classify the "beginning", and the sand being kicked and flying away as the "end". But the event truly not ended, because the sand continues to fly and react to another substances, which react with others in an infinite chain. That's why people make jokes about going back in time, stepping in a bug and change the course of history! Anyway, things just happen in their proper instants of time, the "begin" and "end" part is our doing.
Have a nice day.