In the Meno, Socrates makes the sensible observation that in order for us to ask how virtue is acquired, we must first of all know what virtue is. The hermeneutic circle problem immediately pierces into the discussion, which is to say that in order to enquire as to what virtue is we must already know something about virtue. There is some tacit understanding of "X" built into our enquiries, resulting from the requirement of some form of knowledge or understanding (this distinction will be important) of "X" in order that we are enabled to intend to acquire a deeper or more explicit grasp of it.
The Elenchus, or Socratic method, attempts to draw these understandings out of Socrates’ interlocutors. At this point it’ll be helpful to reiterate that Socrates holds to an epistemic interpretation referred to as anamnesis, which means loss of forgetfulness, or remembering. The idea which Socrates’ investigations are based on is that we have a priori knowledge of what virtue is, and that by asking people questions and cross-examining contradictions which they have in their beliefs, we can approximate closer to what a virtue is.
In relying on non-contradiction in this way Socrates is holding a moral objectivist stance – believing that there is an answer which we all know in some sense, as to what virtue is, and that contradictions will reveal beliefs which cannot both be true simultaneously, and in this way bring us closer to the proper definition (assuming that the law of non-contradiction is ontologically sound). As noted in the Irwin commentary, Socrates must also assume that the judgements made by the interlocutors are sound judgements – which is to say that their judgements capture or approximate what is real (the Greek understanding of truth is “aletheia” which literally means uncovering or disclosing – there is no representationalist account of truth, we do not “create” truth as in subjectivism, but bring it out of concealment).
However, if we have two contradictory definitions of virtue, we’ll call them A and ¬A, both of which are in similar epistemic standing (which is to say that the reasons for doing away with one are as good as the reasons for doing away with the other) – which one are we to assume? What is there to privilege one definition over the other? If we can find no reason, then we cannot formulate an objective definition of virtue.
The term virtue itself originates from “virtutem” which means excellence, moral strength, or value. But what makes something morally sound, excellent, or valuable? We normally consider these words as casting light on a region of human experience vulnerable to the caprice of “subjectivity”. Morality itself is a contingency of an historical sort: the ethics of an era are essentially caught within that era and whatever the projected aim of a given culture, the morality will reflect and guide its conduct. Chinese Legalism, for example, considers human beings as evil and prescribes the use of laws to control them. More liberal societies, however, will put more trust in the conduct of individuals, encouraging free market capitalism and so on.
Anthropologist Pierre Bourdieu notes that social conduct is learned through imitation of bodily gestures, and crystallised firstly in the modes of conduct which engage parent and child. These gestures are not always conceptually explicit, and cannot be taught by manner of textbook and powerpoint. They persist in the pre-theoretical sphere. It’s not important to get into too much depth with this point, and will be sufficient to understand, simply, that the projected aims of a culture inform its moral conduct and therefore what it is to be virtuous.
To return to the hermeneutic circle I mentioned earlier, Socrates asked how we can enquire into what virtue is without having an a priori understanding of it. Heidegger was also inspired by this particular problem in the Meno, and points out that our theoretical knowledge, which is to say our explicit, conceptual definition of what things are, is something wholly different to the kind of a priori knowledge both philosophers envisage. I expect Socrates himself would have granted his ascent to this point, but how the two philosophers consider the a priori is sharply different.
The Irwin commentary again illumines further similarities between Heidegger and Socrates in noting that Socrates believed that speaking of particular instances of virtue do not show knowledge of what a given virtue is. The distinction he draws is one Heidegger elsewhere draws himself, between the ontic (for Socrates this is the particular instance of a virtue, e.g. braveness in combat) and ontological (what a virtue is, the a priori understanding of, e.g. bravery).
Now, both philosophers see truth as some form of uncovering what is already there (“aletheia” in Heidegger’s borrowed Greek terminology, or “anamnesis” in the Socratic paradigm) which distinguishes truth from a subjectivist account, believing we bring out of concealment something concealed but nonetheless existent, rather than bringing things into existence. However, where they differ is that Heidegger believes our pre-theoretical knowledge of what virtue is originates in our socialised mode of being in the world. To put this simply, our tacit (a priori) understanding of what virtue is, which allows us to make this idea of virtue explicit, is not something given outside of human conduct but something which is essentially grounded by it. The Elenchus may reveal, through use of non-contradiction, the definition of virtue which stands for a historically grounded culture and its mode of being – but contrary to moral objectivism, will not provide a definition of morality which will stand for all possible historical cultures.
A further point of departure for the two is that Socrates believes someone must possess theoretical knowledge of what it is to be virtuous, in order to be virtuous at all. Heidegger, on the other hands, speaks of levels of understanding which cannot (and often should not) be made explicitly conscious. To act virtuously under an a priori understanding of virtue, without mental content correlative to the intentional act, is to act virtuously nonetheless (on my reading, anyhow). However, Socrates sees this as merely "accidental" virtue - as we must possess explicit theoretical knowledge that we are acting virtuously in order to count as behaving so. I believe Socrates is prejudicing explicit knowledge over unconscious understanding here, and is doing so without justification.
The specific conduct of one considered virtuous, or acting morally, depends upon a contingent pre-theoretical understanding of what virtue is (or "must be" for that culture) but is otherwise meaningless outside of that nexus. While Heidegger does not wish to demote aletheia to an insipid subjectivism, it does not follow from this that an understanding of the Elenchus in light of hermeneutic phenomenology implies that virtue is something which exists over and above this nexus. Virtue is grounded in social conduct, and so it is only this which is revealed through Socrates’ cross-examinations, and not some approximation to a concrete definition of what is virtuous.