The TV series Stargate is a rather good sci-fi series. However, there are certain logical problems, plot holes and such in the universe of the series, many of which are illogical or contradictory.
It is stated in many episodes that when dialing the gate, the last symbol in the address must be the symbol for the current location of the calling gate. This presents a multitude of logical problems:
Firstly, the amount of different symbols is very limited. It is stated in some episodes that there are only 38 symbols in the dialing device. It's not explained how each stargate location can have their own symbol, distinct from the others.
One possible explanation is that the each dialing device has one unique symbol, distinct from any other dialing device, and that's the last symbol which must be entered. However, it has been shown in most episodes that the gate itself has the same symbols as the dialing device, and thus the gate would also have to have this unique symbol, which would tie it to a specific location. However, it has been shown several times that gates can be moved to completely different parts of the galaxy and used there, and that the last symbol changes accordingly (eg. when a gate is brought from another star system to Earth, the last dialed symbol is changed to that of Earth).
It has also be shown in some episodes that gates can be used with dialing devices other than the original for that gate. This would indicate that all dialing devices and gates have the same symbols and are not tied together. Thus this gives only 38 possible "home symbols".
In one episode a gate was aboard a spaceship, and when dialing was attempted, it failed. It was explained that it was because the gate had been moved too far from its "home location" and thus the last symbol wouldn't work. Thus the last symbol is clearly linked to the gate's physical location and not just a symbol specific to that gate.
This also poses the question why the last symbol is necessary at all (unless it's some kind of security measure, ie. you have to know where you are in order to use the gate).
The episode "Solitudes" badly contradicts this "home symbol" plot. O'Neill and Carter end up in an unknown location, where there's a stargate, and they try to dial Earth, unsuccessfully. No mention is made whatsoever about Carter not knowing the last symbol for dialing, as if Earth's address was a fixed set of symbols not depending on the location of the calling gate. It is stated at the end that dialing the gate didn't succeed because the destination gate was too close (in the same planet). Again, no mention whatsoever about the last symbol being wrong. It seems that the writers just forgot about this plot device.
Actually this episode is much more logical than any of the other episodes dealing with the last symbol. It makes sense that the address to a certain gate is fixed, with no varying last symbol depending on the caller's location. Not having the last-symbol-plot-device removes most of the inconsistencies that having it brings. (Why the gate in the starship couldn't be dialed could be explained eg. by saying that the wormhole cannot form when the gate is in hyperspace.)
In many episodes stargates seem to somehow magically know what their users want it to do, specifically related to when the gate should close the connection. Often someone dials the gate, which opens the connection, and then the connection stays open for as long as it takes for the people to go through, and after the last person has gone through, it takes but a few seconds for the connection to close.
What is even stranger is that in at least one episode the gate closes right after a person being chased goes through, just before the chasers get to the gate (the chaser jumps towards the gate, but it closes the connection and he ends up just jumping through the empty ring).
Exactly how does the gate know when to close the connection?
In some episodes it's stated that the connection will remain open for as long as there's something being dematerialized, or even if there is a radio transmission going through. However, this doesn't explain how the gate remains open between dialing the address and the first person going through. Even if there was a safety mechanism which keeps the gate open until at least something goes through, it doesn't explain why it keeps open after the first person goes through even though a second person has still not even touched the event horizon, and why it closes just seconds after the last person goes through.
Even if it sensed that there are people moving towards the gate and consecuently keeps the connection open, the episodes where the gate closes before any chasers get through heavily contradict this.
Also the radio-signal-keeping-the-gate-open explanation contradicts common sense because, at least on the Earth side, there are always radio signals going on air all the time. It has been shown at least in a few episodes that the Earth gate closes itself without human intervention after the last traveller has gone through, even though radio signals abound.
This may actually not be a plothole at all because, perhaps a bit surprisingly, 6 symbols from a group of 38 may actually be enough to pinpoint almost any star with a habitable planet in our galaxy.
Apparently current estimates on the number of stars in our galaxy is approximately 200 000 millions of stars. Only a small fraction of these stars have conditions for habitable planets, mostly on the outer edge of the galaxy.
A combination of 6 symbols from a set of 38 gives 3 010 936 384 possible combinations (which is about 1.5% of the total number of stars). If we further assume that all stars with conditions for supporting habitable planets are located inside a relatively thin torus shape on the edge of the galaxy and that the ancients used the symbols as toroidal coordinates inside this shape, it may well be that the number of combinations is more than enough to pinpoint any star location inside it with enough accuracy. (AFAIK it's not stated in the series how the symbols map to galaxy coordinates, so this could be conceivable.)
If we go further and forget the last-symbol-is-home plot device (which is full of contradictions anyways, as stated earlier), the accuracy of the addresses increases even further, to 114 415 582 592 (about 57% of all stars in our galaxy).
The eight-symbol addresses needed to dial other galaxies might be plausible too, even though apparently only one symbol is used to pinpoint the destination galaxy. This gives 38 possible galaxies. It may well be that the ancients simply encoded (at most) 38 of the closest galaxies into the stargates, which would be more than enough considering that they probably didn't visit any more galaxies than that. (And who knows, perhaps addresses with even more symbols exist...)
Of course given that star coordinates need to be adjusted for stellar drift, this "galaxy coordinate" symbol must somehow be independent of the current location of that galaxy. This could be explained by something not seen in the series.
On the other hand, an episode of Stargate Atlantis contradicts this idea because it establishes that it's possible to dial to a stargate even if it's located between galaxies. If this was indeed so, then the number of symbol combinations would certainly not be enough even if 8 symbols were used (and even if there was no "home location" symbol), at least not unless the minimum required distance between gates would be enormous.
(On the other hand, one could argue that the gates which form the "bridge" between the two galaxies have been reprogrammed for that specific task and do not use the same address system as regular stargages.)
Contradictory travelling times through the gate are given in different episodes. In one episode Sgt. Harriman counts down the arrival of a MALP to the other side of the wormhole, and it takes several seconds. In a much later episode Carter states that the travelling always takes exactly 0.2 seconds. The on-screen animation of the travelling always takes several seconds.
Many episodes seem to imply that when a gate starts dialing, the target gate starts spinning at the same time. It's unexplained how the target gate knows it's being called before the entire address has been entered. Logically at least the first 6 symbols must be entered before the target gate can be decided. Somehow, however, gates seem to be able to magically know they are being called before this.
One possible explanation is that the target gate actually starts spinning only after the 6th symbol has been entered. However, it takes some time for the target gate to spin through all the symbols, and there's no delay in the forming of the wormhole, which happens immediately after the last symbol has been entered in the calling gate. One would assume that the wormhole is created on both ends simultaneously, without a long delay. After all, people often enter the gate very soon after it has opened, so we can assume that it has also opened on the target gate as well.
In several episodes it's stated that Sam Carter and many others believe that there is an infinite amount of parallel timelines, and that any modification to a timeline simply causes another parallel timeline to form (implying that the timeline which caused the tampering is not affected).
This is somewhat contradicted by an episode where a future Sam Carter suggests sending a message to the past in order to stop the past (viewer's present) SGC from making the mistake that caused those future events.
If the infinite-timelines theory was true, then sending that message would have no effect whatsoever on that Carter's timeline. Sending the message would basically be charity for some timelines, but would be useless for that Carter's own timeline.
However, the episode seems to imply that sending the message would effectively change the timeline and make that future to cease to exist.
In a way this actually is more desirable because it makes time-travelling more interesting storywise. Of course time-travelling inside one single timeline breaks the law of conservation of energy (I'll just skip explaining why this is so), but this can be logically explained: It could be stated that the gate simply consumes additional energy when it needs to make the kind of time travelling which would otherwise break conservation of energy happen.
(Curiously, this kind of time-travelling could be used to, for example, use the gate to duplicate objects, eg. valuables. I'll leave the reason for this for the reader as a thinking exercise. Physicswise this can just be explained with the gate consuming the extra energy necessary for this, as stated above.)
A plot influencing many episodes is that at some point SGC's gate is destroyed and they have to rent the one the Russians have. The government pays money and share technology and whatnot just to keep the gate.
Given that at some point SGC owns spaceships capable of travelling between star systems and more than big enough to transport stargates, and given that stargates can be moved and used from anywhere, why couldn't they simply "borrow" an abandoned stargate from an uninhabited planet and return the Russian gate? This would save them money and lessen their dependency on the Russians.
Although this was a deliberate decision by the makers (in order to simplify things) and an admitted illogical element of the series, it still bothers me a bit.
Most people in the known universe (including people from other galaxies) speak modern English as their native language, or at least understand it and speak it fluently. This is so even for people who were brought to a planet thousands of years ago (when English didn't even exist) and have been completely isolated there ever since.
In one episode SG1 encounters a population which seems to be based on a culture of the Middle Ages on Earth, possibly from England. While it might be conceivable that they spoke English of that period when they were transported to that planet, it would still be highly unlikely that their spoken English developed in the same way as on Earth. In the episode they speak perfect modern English, without even the slightest of accents.
Yet, regardless of this inexplicable "universal language", some alien races speak other languages too, most prominently the goa'uld and the ancients, although they, too, understand and speak fluently modern English. However, other than them, it seems that no other culture in the known universe speaks any other language than English.
Quite curiously, in an episode where Apophis (the symbiote) dies, his host doesn't speak English at all, but only ancient Egyptian. This makes it a bit inconsistent why in some cases someone doesn't understand English while in most other cases even people who have been completely isolated for thousands of years do.
Quite inconsistently, too, it seems that the only place in the universe where different languages have evolved over the millenia is Earth, where people speak different languages, while in the rest of the universe modern English is basically the only spoken language.