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Dartmouth German
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The Present Perfect Tense in English:

The present perfect tense describes a past event that has present tense implications (compare the simple past "I cooked twice this week" with the present perfect: "I have cooked twice this week" - the former implies that that's all the cooking I'm going to do, while the latter suggests that I might cook more).

English forms this tense with a combination of the auxiliary verb "to have" and a past participle: "I have noticed that," "She has gone there twice," etc. The name "present perfect" reflects the fact that the auxiliary verb, "to have," is conjugated in its present-tense forms.

Some verbs long ago used "to be" as the auxiliary, and we find vestiges of this structure in biblical English or old nursery rhymes:

The Lord is risen.
For He is come.
Elsie Marley is grown so fine, she won't get up to feed the swine, but lies in bed till eight or nine.

Weak and strong verbs form past participles differently.

Weak verbs add "-ed": "looked," "wanted," "worked," etc.
Strong, or irregular, verbs generally (but not always) change the vowel in the stressed syllable and typically (but not always) add "-n" or "-en": "risen," "written," "eaten," "sung," "slept," "brought," etc.

Using the Present Perfect Tense (das Perfekt) in German:

In German, as in English, the present perfect differs from the simple past, in that it describes past events that have present implications. German speakers are not always careful in making this distinction, however. Indeed, they sometimes even mix the two tenses indiscriminately.

Even more important: in colloquial conversation, Germans use the present perfect almost exclusively. Indeed, many dialects do not even have a simple past, which is thus mostly reserved for written narrations. Colloquial accounts are in the present perfect: "Ich bin nach Hause gegangen und habe meinem Mann gesagt...." There are a few exceptions, such as the verb "sein" and the modal auxiliaries. It is quite common to use the less complex "ich war da" instead or "ich bin da gewesen" or "sie konnte ihn sehen" rather than "sie hat ihn sehen können." Even in ordinary speech it is more usual to say "ich musste einen Arzt rufen lassen" than "ich habe einen Arzt rufen lassen müssen." "Er hatte einen Hund" is also possible in place of "Er hat einen Hunde gehabt."

Forming the Present Perfect Tense in German:

Weak verbs typically form the past participle by adding the prefix "ge-" and a suffix of "-t" or "-et" to the stem:

Ich habe das gesagt. I said that.
Sie hat gut gespielt. She played well.
Wir haben schwer gearbeitet. We worked hard.
Ich habe nichts gehört. I didn't hear anything.

If the verb has a separable prefix, the "ge" becomes an infix; it is placed between the prefix and the stem:

Wir haben die Tür zugemacht. We closed the door.
Ich habe eingekauft. I went shopping.

If the verb ends in -ieren, there is no ge- added:

Das hat gut funktioniert. That worked well.
Sie hat Physik studiert. She studied physics.
Hast du auch die Küche renoviert? Did you renovate the kitchen, too?

If the verb has an inseparable prefix, that replaces the "ge":

Was hast du ihnen erzählt? What did you tell them?
Habt ihr viel Geld dafür bezahlt? Did y'all pay a lot of money for that?
Der Wagen hat mir gehört. The car belonged to me.

Strong verbs:

The "irregular strong verbs," including the modal auxiliaries when they are not associated with another verb in the infinitive, add the suffix "-t" or "-et" to a (usually) changed stem:

Ich habe das nicht gewollt. I didn't want that.
Als Kind habe ich gut Chinesisch gekonnt. As a child I could speak Chinese well.
Sie hat das nicht gewusst. She didn't know that.
Was hast du mir gebracht? What did you bring me?

The remaining strong verbs add "-en" to a stem that may or may not be changed.

Ich habe meinen Hut gefunden. I found my hat.
Du hast zu schnell gesprochen. You spoke too fast.
Haben Sie gut geschlafen? Did you sleep well?
Sie hat mein Buch nicht gelesen. She didn't read my book.

These strong verbs fall into distinct categories. Here is a list of the most common strong verbs, arranged in those groups.

The rules about the prefix "ge-" remain the same:

Was hast du mitgebracht? What did you bring along?
Das Konzert hat schon begonnen. The concert has already begun.
Sie hat schon alles aufgegessen. She has already eaten up everything.
Die Kinder haben den ganzen Tag ferngesehen. The children watched TV all day.
Wir haben etwas anderes vorgehabt. We planned to do something else.

The Auxiliary Verb: Most verbs, as in the examples above, take "haben," but some require "sein":

Wann bist du nach Hause gekommen? When did you come home?
Wir sind ins Kino gegangen. We went to the movies.
Seid ihr geflogen oder gefahren? Did y'all fly or drive?
Er ist alt geworden. He's gotten old.
Sie sind in der Stadt geblieben. They stayed in the city.
Die Musik ist sehr laut gewesen. The music was very loud.

  New Mercedes-Benz M-Class Rolls Off Assembly Line
The new Mercedes-Benz M-Class rolled off the assembly line today in Tuscaloosa Alabama (USA).
M-Class: New SUV perspectives for Daimler

The verbs that take "sein" are mostly predictable on the basis of their meaning. They must satisfy two conditions: 1) they must be intransitive; 2) they must indicate a change of position or of condition. In the example "Wir sind nach Hause gegangen," the verb "gehen" 1) takes no direct object and 2) describes motion from one place to another. The same is true with:

Sie ist spät aufgestanden. She got up late.
Er ist gestern angekommen. He arrived yesterday.
Ich bin langsam gelaufen. I ran (or walked) slowly.

Examples of a change of condition:

Ihr Ring ist schon grün geworden. Her ring has already turned green.
Er ist gestern gestorben. He died yesterday.
Die Pflanze ist schnell gewachsen. The plant grew fast.
Ich bin sofort eingeschlafen. I fell asleep immediately.

Some Other Wrinkles:

1) There are two obvious exceptions to these rules: "bleiben" and "sein". While they are intransitive (or, from another point of view, take the nominative case), they clearly do not show a change of position or condition. In fact, they specifically mean not to make such a change. Nevertheless, they take "sein." "Folgen" might also be a surprise, unless we consider its use of the dative.

Er ist zu Hause geblieben. He stayed home.
Er ist ein Junge geblieben. He remained a boy.
Es ist immer so gewesen. It was always that way.
Der Mann ist mir nach Hause gefolgt. The man followed me home.

2) With some verbs, the context determines whether or not they take "sein." "Fahren," for example has two meanings: a) to drive in the sense of riding in a vehicle ("Wir sind nach Berlin gefahren") and b) to operate a vehicle ("Ich habe deinen Wagen gefahren"). In the second case, "haben" is called for because "fahren" is transitive. Other examples:

Wir sind nach Madrid geflogen. We flew to Madrid.
Der Pilot hat das Flugzeug allein geflogen. The pilot flew the plane alone.
Wir sind um zwei gelandet. We landed at two.
Der Pilot hat das Flugzeug um zwei gelandet. The pilot landed the plane at two.

3) With a verb like "fahren," those two meanings are sufficiently distinct. In some other instances, a certain amount of good will is called for. "Gehen," for example, can actually, though rarely, take an object, but one still says, "Er ist seinen eigenen Weg gegangen" ("He went his own way"). And one normally says, "Ich bin einen Marathon gelaufen" - although some Germans would differ and insist on "haben."

  Ran Holding His Brother's Hand
The recreational athlete Thomas Steinbrenner from Heidelberg ran the Berlin Marathon today holding his brother's hand. It was his fourth marathon in Berlin.

4) The movement implied in "tanzen" is not sufficient for "sein." Hence: "Ich habe nur mit ihm getanzt." - Although: "Ich bin mit ihm ins nächste Zimmer getanzt" ("I danced with him into the next room" [think tango]). "Schwimmen," in contrast, most often takes "sein," even if it's just doing laps: "Ich bin heute nur kurz geschwommen" ("I swam today just for a short time").

5) "Stehen" and "sitzen" obviously do not meet the criterion of motion, but Southern Germans, to the horror of Northerners, typically say "Ich bin gestanden" or "Ich bin gesessen." Foreigners who use this construction get their knuckles rapped.

Double Infinitives:

The modal auxiliaries behave differently when paired with an infinitive. The present perfect form of "Ich kann Deutsch" is "Ich habe Deutsch gekonnt." But "Ich kann Deutsch sprechen" becomes "Ich habe Deutsch sprechen können." Other examples:

Ich habe das nicht wissen können. I couldn't know that.
Sie hat das nicht machen müssen. She didn't have to do that.
Wir haben ihm nicht schreiben dürfen. We weren't allowed to write to him.

The same is true of verbs of perception that take an infinitive without "zu." "Sie hört ihn singen" becomes "Sie hat ihn singen hören." Other examples:

Ich habe sie schwimmen sehen. I saw her swimming.
also possible: Ich habe sie schwimmen gesehen.
Hast du ihn kommen hören? Did you hear him coming?
also possible: Hast du ihn kommen gehört?

Two other verbs, lassen and helfen, also form double infinitives

Wir haben ein Haus bauen lassen. We had a house built.
Ich habe ihr kochen helfen. I helped her cook
also possible: Ich habe ihr kochen geholfen.

In these cases, the double infinitive remains in the final position in dependent clauses, and the "haben" slips into the second-the-last place:

Bist du sicher, dass sie das Buch hat lesen können?
Are you sure that she was able to read the book?
Wir sind nach Hause gegangen, weil wir keine Karten haben kaufen können.
We went home because we couldn't buy any tickets.
Es ist schade, dass du ihn nie hast singen hören.
It's too bad that you've never heard him sing.

In the above examples, the modal was put into a perfect tense. As in English, the modal can have a different meaning when combined with another verb that is in the past. Note the following distinctions (See also special meanings of the modal auxiliaries):

Sie hat das sagen dürfen. She was allowed to say that.
Sie darf das gesagt haben. She may have said that.
Er hat mir einen Brief schreiben können. He was able to write me a letter.
Er kann mir einen Brief geschrieben haben. He may have written me a letter.
Sie haben mich nach Hause tragen müssen. They had to carry me home.
Sie müssen mich nach Hause getragen haben. They must have carried me home.
Ihr habt mir helfen sollen. You were supposed to help me.
Ihr sollt mir geholfen haben. You are supposed to have helped me.
Er hat es finden wollen. He wanted to find it.
Er will es gefunden haben. He claims to have found it.