"We are saying."
In present tense, you add "-iyor" to the verb stem. If the verb ends in a vowel, you erase that completely. For example: "izlemek" (to watch). The root is "izle". It ends in a vowel, so you drop it: "İzl-". Now add "-iyor" → İzliyor.
Same thing with "demek". The root is "de-". Omit the vowel: "d-". Add iyor → diyor.
One more thing to add:
If the verb ends with a vowel but has more than one syllable, the 4-way in the ending -Iyor is created by referencing to the vowel that is not erased. So..
Söyle- (this is root)
Söyl- (you erased the last vowel)
Söylüyor. (-ü comes from ö according to the 4-way rule)
This sentence has the same problem as another, similar one, where I already commented: "We are saying" doesn't make any sense as a complete sentence, and I think the same is true for the Turkish sentence. Therefore the full stop is misleading. In Duolingo it is usually not present in sentence fragments (such as individual words), so it should not be present here, either. Otherwise we are bound to draw incorrect conclusions such as: "'We are saying.' doesn't make sense as a full sentence, so it must mean something different in Turkish. Maybe demek can also mean to talk. Let's try 'We are talking.'"
Or maybe it's a full sentence in Turkish, after all, and the proper translation is "We are saying something."?
From what I have learned so far:
- Demek is used for direct speech, as in "He said 'It's true!'"
- Söylemek is used for indirect speech, as in "He said it was true."
It appears that otherwise they are fairly interchangeable. English has similar redundancies, though not in exactly the same way. Just think of say, speak, and tell.
Well, the different ways in which they are used to report speech are certainly a clue. Also, from the Wiktionary entries we can learn that tell is an alternative translation for söylemek but not for demek and that both words derive from Old Turkic words with different meanings:
- demek from say, call
- söylemek from speak, promise.
All of this seems to support the idea that demek is slightly more neutral and söylemek slightly more formal.