The Ancient Greek infinitive is a non-finite verb form, sometimes called a verb mood, with no endings for person or number, but it is (unlike in Modern English) inflected for tense and voice (for a general introduction in the grammatical formation and the morphology of the Ancient Greek infinitive see here and for further information see these tables).
It is used mainly to express acts, situations and in general "states of affairs" that are depended on another verb form, usually a finite one.
It is a non declinable nominal verb form equivalent to a noun, and expresses the verbal notion abstractly; used as a noun in its main uses, it has many properties of it, as it will be seen below, yet it differs from it in some respects:
Ancient Greek has both (a) the infinitive with the article (articular infinitive), for example τὸ ἀδικεῖν "doing wrong, wrong-doing" and (b) the infinitive without the article, for example ἀδικεῖν "to do wrong".
The articular infinitive corresponds to a cognate verbal noun (in singular number only). It is preceded by the neuter singular article (τό, τοῦ, τῷ, τό) and has the character and function of both a noun and a verbal form. It can be used in any case (nominative, genitive, dative, accusative) and thus participate in a construction just like any other noun: it can be subject, object (direct or indirect), predicative expression (rarely), or it may also serve as an apposition; it may have an adnominal (e.g. to be in a genitive construction as a possessive or objective genitive etc.) or an adverbial use (e.g. it can form a genitive that denotes cause etc.); it may form an exclamation (in poetry); it can also be the complement (object) of a preposition in any oblique case and denote many adverbial relations; finally, if in the genitive case, it can denote purpose, oftener a negative one.
In all the preceding passages the articular infinitive is in the present tense stem; yet this is by no means a rule, since it can be used in any tense stem, denoting a variety of aspectual differences (For more details see below the discussion about the present and aorist dynamic infinitive).
The infinitive without the article is of two sorts and has two discrete uses: the dynamic infinitive and the declarative infinitive. Traditionally they are said to be used not in indirect discourse and in indirect discourse respectively, yet this terminology is misleading; for infinitives of both sorts may be used in indirect discourse transformations (for example one may say (a) "I said that he will undertake an expedition" or (b) "I advised him to undertake/that he should undertake an expedition", where indirect discourse, one way and another, is employed: direct discourse for (a) "He will undertake...", and for (b) "Undertake/you should undertake...").
A so-called dynamic infinitive may be governed by verbs of will or desire to do something (ἐθέλω or βούλομαι "to be willing, wish to", εὔχομαι "pray, wish for", κατεύχομαι "pray against, imprecate curse to", αἱροῦμαι "choose, prefer to", μέλλω "to be about to, or: delay to", κελεύω "urge, command to", ἐπιτάσσω "order to", ψηφίζομαι "vote to", ἐῶ "allow to", δέομαι "beg to" etc.), verbs of will or desire not to do anything (δέδοικα/δέδια "fear to", φοβοῦμαι "be afraid to", ἀπέχομαι "abstain from doing", αἰσχύνομαι "be ashamed to", ἀπαγορεύω "forbid to", κωλύω "hinder, prevent" etc.) and verbs or verbal expressions denoting ability, fitness, necessity, capacity, etc. (δύναμαι, ἔχω "be able to", ἐπίσταμαι, γιγνώσκω "know how to", μανθάνω "learn to", δυνατὸς εἰμί, ἱκανὸς εἰμί "I am able to", δίκαιον ἐστί "it is fair/right to", ἀνάγκη ἐστί "it is necessary to", ὥρα ἐστί "it is time to" etc.). It can also be found after adjectives (and sometimes derived adverbs) of kindred meaning (δεινός "skillful", δυνατός "able", οἷός τε "able", ἱκανός "sufficient, capable" etc.). It stands as the object (direct or indirect) of such verbs or verbal expressions, or it serves as the subject if the verb/the verbal expression is used impersonally; it also defines the meaning of an adjective almost as an accusative of respect. An infinitive of this kind denotes only aspect or stage of action, not actual tense, and can be in any tense stem (mostly in the present and aorist (see also here), the perfect being rare enough) except the future one; only the verb μέλλω "I am about to" may exceptionally take a dynamic future infinitive.
The difference between the present and the aorist infinitive of this sort is aspect or stage of action, not the tense —despite their tense stem, such infinitives always have a future reference, because of the volitive meaning of their governing verb. More specifically, an infinitive in the present verb stem lays stress on "the process or course of the state of affairs", and in many cases has "an immediative" semantic force, while an infinitive in the aorist verb stem lays stress "on the completion of the state of affairs, expressing a well-defined or well-delineated state of affairs".
Analogous aspectual distinctions between the present and aorist verbal stem are present also in the use of finite moods as the imperative and the subjunctive and even the optative of wishes in independent clauses. So, in cases as those presented in the following examples, a dynamic infinitive somehow recalls a corresponding finite mood expressing will or desire, pray or curse, exhortation or prohibition etc. and indirect discourse is from one aspect employed:
A so-called declarative infinitive (see also declarative sentence) is mostly used in connexion with verbs (or verbal expressions) of saying, thinking and (sometimes) perceiving such as λέγω, φημί, ἀποκρίνομαι, ὑπισχνοῦμαι, ὁμολογῶ, ἀκούω, ὁρῶ etc. and it is usually used in oratio obliqua (in indirect speech or indirect discourse). The latter means that it represents a corresponding finite verb form of the oratio recta (of the direct speech or discourse), thus a declarative infinitive denotes both tense and aspect or stage of action. But the present infinitive represents either a present indicative or an imperfect one, and a perfect infinitive either a perfect indicative or a pluperfect one. A declarative infinitive with the particle ἂν is also the representative of a potential indicative or potential optative of the corresponding tense.
Verbs that usually have a future reference, such as ὄμνυμι "swear", ὑπισχνοῦμαι "promise", ἐλπίζω "expect, hope", ἀπειλέω "threaten", προσδοκάω "expect" etc., either take the declarative infinitive (mostly the future, but less often some of them also take the present, aorist or perfect infinitive, even the infinitive with the particle ἄν representing a potential optative or indicative), and in this case indirect discourse is employed, or they are followed by the dynamic aorist (less often the present) infinitive, and they are constructed just like any verb of will, desire etc. The same constructional alternation is available in English (declarative content clause -a that clause- or to-infinitive), as shown below.
For the difference between the present and aorist dynamic infinitive see the discussion in the above section. Yet in the last two examples another reading is also possible, considering ἀποδιδόναι and ἀποδοῦναι to be present and aorist declarative infinitive respectively: "I swear that I give (always, or in any relevant situation etc.) the money back. I swear that I gave the money back."
The ("dynamic") infinitive is used instead of the indicative mood, with substantial difference in meaning, in certain subordinate clauses introduced by specific conjunctions: ὥστε (ὡς) "so as to, so that", πρίν (πρόσθεν... ἤ) "before" or "until" and relative adjectives introducing relative clauses of result, such as ὅσος "so much as enough to", οἷος "of such a short as to", ὃς or ὅστις "(so...) that he could", in clauses introduced by the prepositional phrases ἐφ' ᾧ or ἐφ' ᾧτε or with ὥστε "with the proviso that".
Note: a "declarative" infinitive is sometimes the mood of subordinated clauses in indirect speech, instead of a corresponding indicative (either a realis or conditional irrealis one) or optative mood, in modal assimilation to the main infinitive used to represent the independent clause of the direct speech; so after relative, temporal or conditional conjunctions such as: ὃς "who" or ὅστις "whoever", ἐπεὶ or ἐπειδή "since, when", ὅτε "when", εἰ "if" etc. An example:
Here, the main infinitives, those directly depended on the finite verb ἔφη, namely πορεύεσθαι and ἀφικνεῖσθαι, attractivelly affect the mood of the embedded clauses introduced by ἐπειδὴ, a temporal conjunction, and ἐν ᾧ, a relative prepositional phrase.
In general, Greek is a pro drop language or a null-subject language: it does not have to express the (always in nominative case) subject of a finite verb form (either pronoun or noun), unless it is communicatively or syntactically important (e.g. when emphasis and/or contrast is intended etc.). Concerning infinitives, no matter of which type, either articulated or not, and also either of the dynamic or declarative use, the following can be said as a general introduction to the infinitival syntax (:case rules for the infinitival subject):
These three main constructions available are desctribed in some detail in the sections below.
The construction where an accusative noun or pronoun functions as the subject of an infinitive is called accusative and infinitive (See also the homonymous Latin construction accusativus cum infinitivo (ACI), which is the rule -in indirect speech- even in cases where verb and infinitive have co-referential subjects). This construction can be used as an indirect speech mechanism, in many instances interchangeable with a complementary declarative clause introduced by "ὅτι/"ὡς" (or a supplementary participle). But with some verbs (normally with verbs of thinking, as νομίζω, οἴομαι, ἡγέομαι, δοκέω etc., with the verb φημί "say, affirm, assert", with verbs denoting hope, oath or promise, such as ἐλπίζω "hope", ὄμνυμι "swear", ὑπισχνοῦμαι "promise", etc.) the infinitival construction is the rule in classical Greek. Yet it can be also in use with any infinitival use, no matter whether indirect speech is involved or not. In the following examples the infinitival clause is put in square brackets :
Some actual examples from classic Greek literature:
Oratio recta/Direct speech would have been: τοὺς πονηροτάτους καὶ ἐξαγίστους ὀνομαζομένους αἱNOM συμφοραὶNOM σωφρονίζουσινFIN. "The mishaps chasten those called utterly wicked and ungodly". (Articulated substantive as subject of the finite verb would have been put in nominative case)
Oratio recta/Direct speech would have been: ἡNOM ἡμετέρα φύσιςNOM ἱκανωτέραNOM ἐστὶFIN τῆς ὑπὸ τῶν θεῶν προκριθείσης. "Our nature is more competent than the one chosen by the gods as best". (Articulated substantive -subject of the finite verb- and predicate adjective both in nominative case)
And here is an example where no indirect speech is involved:
This construction, accusative and infinitive, is also always in place when the main verb is an impersonal one or an impersonal verbal expression, and the infinitival clause functions as its subject (here also there is no indirect speech). Of course, in such cases the infinitive has a subject of its own. An example:
When the subject of the infinitive is identical (coreferential) with the subject of the governing verb, then normally it is omitted and understood in the nominative case. The phenomenon is traditionally understood to be some kind of case attraction  (for a modern perspective and relevant modern terminology see also big PRO and little pro and control constructions). In the following examples infinitival clauses are bracketed ; coreferent items are indexed by means of a subscripted "i".
Πέρσης is a predicate noun in the nominative, showing case agreement with an understood and omitted pronoun (Here we are dealing with coreferential proi and PROi).
Note: there are certain cases where the subject of the infinitive, whether of the declarative or the dynamic type, is put in accusative case, even though it is co-referent with the subject of the main verb; in this mechanism emphasis or contrast is present. An example:
Here the unemphatic dropped null-subject (if emphatic, a 1st person pronoun ἐγώi NOM should be present) of the main verb is emphatically repeated right after the verb within the infinitival clause in accusative case (ἐμέ, "I"). The meaning is ‘I believe that it is I who have made more money than any other two sophists together – you may choose whoever you like’. The comparative nominal phrase ἢ ἄλλους σύνδυο shows case agreement with ἐμέ.
Here now the subject ἐγώ of the finite verb εἴργασμαι (a perfect indicative) is emphatically uttered in nominative case; the second part of the comparison, ἢ ἄλλοι σύνδυο, agrees with this in nominative case.
When the infinitival subject is coreferent with a word constructed with the governing verb in a higher syntactic level, in other words, when the subject of the infinitive is itself (a second) argument of the governing verb, then it is normally omitted and understood either in the oblique case in which the second argument is put (see also in the previous paragraph the reference to PRO and control structures), or in the accusative as if in an accusative and infinitive construction (but with the accusative noun or pronoun obligatorily suppressed and implied).
In all the above examples the case of the subject of the infinitive is governed by the case requirements of the main verb and "the infinitive is appended as a third argument" (Concerning the second and third examples, in modern linguistic terms we have to do with an object control construction). As fas as the two first are concerned, traditionally this construction is sometimes called (in Latin terminology) dativus cum infinitivo or genitivus cum infinitivo (dative with the infinitive or genitive with the infinitive respectively) and is considered to be a case attraction, the dative or genitive being used instead of a predicate in the accusative: ἄνδρα, ὡς προθυμότατον; see also below.
On the other hand, as it is indicated by predicate adjectives/sunstantives or participial constituents of the infinitival clause, it is not unusual at all for an accusative to be understood and be supplied by context as the subject of the infinitive, as the following examples illustrate. As far as the genitive is concerned, a predicate substantive or a participle normally stands in the accusative while an adjective may stand either in accusative or in genitive case. As far as the dative is concerned, the choice between a word in concord with a dative and an accusative case seems to be laid down by the speaker's/writer's preference.
This construction is obligatory when the infinitive is governed by a participle in any oblique case, more usually an attributive one (and in the nominative also). Here the predicate adjective always shows concord with the case of the leading participle. So an embedded participial clause like φάσκοντες εἶναι σοφοί "claiming that they are wise" or οἱ φάσκοντες εἶναι σοφοί "Those who claim that they are wise" is declined this way -in any of the following word ordering, but in slightly different each time meaning (see topicalization and focusing):
In the above phrasal structuring the predicate adjective σοφοὶ "wise" is always put in the case of its governing participle φάσκοντες "claiming".