Your question is a very good one. Most Turks could not explain it in a hundred years even if they intuitively make the right choices. A learner can acchieve a slight intuition also after diving into the language deep enough, I know it from experience.
While we wait for the masters to clear it allow me to tell you my learner´s view - or maybe mix it a little bit more.
The verbal noun -mA denotes the somewhat abstract concept of an action or its result. -DIK participle is narrative. Your examples are good:
Trenin kalkmasına az kadlı. Little (time) remained for the train´s departure. This uses -MA.
Sorununu çok iyi anladıklarını biliyorum. I know that they understand your problem very well. This uses -DIK.
The first action, the train´s leaving, doesn´t exist anywhere else but as a concept in the speaker´s head. Similarly, -mA infinitive is used when the result of the action is the issue like in
Trenin kalkması beni biraz üzdü. ´The train´s leaving made a little sad.´
Also when we talk about actions that are necessary we see them as abstract (future) concepts in our head.
Gitmem lazım. ´I must go, my going is necessary.´
The place of possessive marked -DIK participle is often with verbs denoting saying, thinking or knowing. What do we say, think or know? Things that have/haven´t actually happened, actions we want to relate, to tell about, to share. Like in your example, jas206: the speaker wants to say he knows what the actual state of affairs is.
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