Three forms of argumentation use the notion of circumstance:

— The fallacy of omission of the relevant circumstances, a criticism addressed to an argumentation.
— The argumentation by the circumstances.
— In the expression “circumstantial ad hominem”, the circumstances alluded to are the characteristics of the person implicated in an ad hominem argument.

1. Fallacy of omission of relevant circumstances

The fallacy of omission of circumstances is sometimes referred to by the Latin label secundum quid fallacy, which abbreviates the phrase a dicto secundum quid ad dictum simpliciter, “from a restricted affirmation to an absolute affirmation”.

Aristotle classifies the fallacy of omission of relevant circumstances as a kind of fallacy “independent of language” (Soph. 4; 165b20; S. Fallacy (2), occurring when an expression is used “absolutely or in a certain respect” (Soph. 5; 166b35):

“If < what is not is the object of an opinion >, then < what is not is >” (ibid.; our emphasis and parenthesis).

What is not is the object of an opinion” is a semantically complete, syntactically integrated utterance, a meaningful unique and complete speech act [1]. All its components are necessary and interdependent; none can be subtracted without altering what the speaker said and meant, and he has only said one thing.
It is not possible to extract from this complete utterance any arbitrarily chosen segment (here, « the object of an opinion”) as long as it makes some sense, and attribute the resulting segment (here, “what is not is”) to the speaker of the former statement.
Such considerations are crucial when it comes to determining what is an elementary well-formed linguistic formula.

Other examples: the specified expression “A is (Place, Time)”, “A is here now” can be reshaped into the corresponding, non-qualified, one “A is (Place)”, “A is here”. Vice versa, the non-specified construction “Peter crossed the street” cannot be specified into “Peter crossed the street yesterday” (which can be non fallaciously reduced to “Peter crossed the street”).
This kind of de-contextualization of a qualified statement may result in irony:

S1:    — The weather is fine! (said in the morning, when the weather is fine).
S2:    — Ah hah! And you said that the weather is fine! (said in the evening, while it is raining).

This fallacy passes over relevant contextual data, treating as an absolute assertion what has been asserted with reservation, in a particular context, with precise reference and intention. This radicalization of assertions and positions makes them very easy to refute.

To be relevant in a methodologically equipped context, the refutation must relate exactly to the expression as used, and take into account all the reservations specifically mentioned. The fallacy is particularly vicious when it pretends that the speaker had fully said and assumed something he or she has only said, in the flux of a dispute, as a concession to the opponent.

Prime Minister: — Our country cannot take in all the misery of the world (S1) but it must take its share (S2).
Opponent : — As Mr. Prime Minister said, we cannot welcome all the misery of the world.

In Goffman’s words, in statement S1 the Prime Minister speaks as an Animator, quoting an unknown Principal, whom he opposes; whereas he speaks as the Principal of S2, taking full responsibility for the content and actions, intentions and consequences of what S2 means, S. Roles.
The opponent forces him to speak as Principal of S1. While the Prime Minister advocates receiving refugees, the opponent, who advocates closing the frontiers, makes an ally of the Prime minister who actually rejected his or her position.

2. Argumentation by the circumstances

Argumentation by the circumstances establishes indirectly the existence of a fact, exploiting peripheral, unnecessary indices of an action that have no real probative value, but nevertheless point to a fact:

Question: — Is he corrupt?
Accuser: — Certainly. He needed money; we have seen him receiving thick envelopes; and yesterday, he bought a brand new car.

In classical terms, the argumentation by circumstances can help to solve a conjectural issue, S. Stasis, such as “did he commit this crime?” (Cicero, Top., XI, 50; p. 82). To answer, one “[looks] for the circumstances that preceded the fact, that accompanied it, that followed it” (Cicero, ibid; XI, 51, p. 83), interpreting “an appointment […] the shadow of a body […] pallor… and other indications of trouble and remorse” (id., XI, 53, p. 83). This is part of the investigatory technique:

He went out murmuring…: this is to argue from what precedes the action; we saw him stealing behind a bush…: that’s what accompanies it. […] a malicious joy, which he endeavored to keep concealed, appeared on his face, mixed with fright: which is what follows.”
Bossuet [1677], p. 140, S. Collections (III)

These observed circumstances are probable natural signs.
Argumentation by the circumstances is a powerful instrument in the arts of suspicion and construction of a culprit.

3. Terminological delicacies

On §53 of the Topics Cicero deals with arguments drawn from “consequences, antecedents, contradictory things [ex consequentibus et antecedentibus et repugnantibus]” (Top., XI, 53: 83).
This paragraph deals with logical antecedence and consequence, involving semantically “necessary” links (id.), referring to questions of a priori and a posteriori reasoning, definition, rules of implication and to the non-contradiction principle.

Bossuet speaks, in connection with the argument by circumstances, of places “derived from what precedes, from what accompanies and what follows [the action], ab antecedentibus, ab adjunctis, a consequentibus” ([1677], p.140). Here, the links of the preceding and following events with the central event are no longer semantical or logical but purely chronological (the change of preposition – ex antecedentibus for the logical consequence and the necessary link vs. ab antecedentibus for temporal anteriority has nothing to do with this distinction).

[1] For example, Empedocles argues that « from nothingness nothing can absolutely come into existence and what is cannot perish. »