Ad ignorantiam argument, Lat. ignorantia, “ignorance”

1. Argumentation from ignorance and legitimacy of doubt

Argumentation from ignorance is defined by Locke as one of the four fundamental forms of argumentation, S. Collections (II):

Secondly, another way that men ordinarily use to drive others, and force them to submit to their judgment, and receive the opinion in debate, is to require the adversary to admit what they allege as a proof or to assign a better. And this I call argumentum ad ignorantiam. ([1690]; Vol. II, p. 410-411)

This argument is considered to be fallacious:

It proves not another man to be in the right way, nor that I ought to take the same way, because I know not a better. (Id., p. 411)

The following dialogue schematizes the situation where S1‘s conclusion relies on the ignorance of S2:

S1_1:   — C, since A.
S2_1:   — This is a bad argument. I do not admit that A proves C.
S1_2:   — Do you have any reason to conclude anything different from C? Do you know a better argument for C?
S2_2:   — Well, no
S1_3:   — Then you have to accept my own proof and my conclusion.

(i) First turn: S1_1 proposes a justified claim C.

(ii) Second turn: S2_1 refuses to ratify the claim C.

(iii) Third turn: S1_2 asks S2 to explain the reasons for his or her doubt. According to the conversational principle which requires justification for non-preferred second turns, S1 is perfectly justified in doing this. S2 could answer:

(a) by presenting objections against the alleged argument, A, or by utterly refuting A;
(b) by constructing a counter-discourse by providing what Locke terms “a better proof”. The text does not tell for what conclusion; so we can therefore assume the following two cases:

(b1) Concluding something different from C;
(b2) Providing “better evidence” for C.

(iv) Fourth turn: S2_2 admits that he or she cannot elaborate anything along the (a), (b1) or (b2) lines.

(v) Fifth turn: S1 may accordingly:

(a) Admit S2’s reluctances, while maintaining his argumentation: “Okay, this is not a very good argument, but it is still interesting, it is even the only one we have”;
(b) Summon S2 to accept his (A, C) argumentation, considering that his partner’s incapacity is a kind of second order proof to add to his former substantial one, A, and so committing an ad ignorantiam fallacy (even if his former argument is, after all, not so bad).

A pure ad ignorantiam fallacy would be based only on the partner’s failure “to assign a better [proof]”. Under conversational circumstances, S2_1 does not ratify S1_1’s turn; normally, this should urge S1 to clarify and elaborate upon his proposal. The crude reaction seems rare: “as you cannot articulate anything against my argumentation, you have to accept it wholesale”.

Seen from S2’s perspective, this situation also seems a little bizarre, a kind of borderline case where S2 has only his or her inner conviction to oppose to an argumentation. Under standard conditions, a conversationalist and a fortiori a dialectician, knows how to elaborate upon a strong inner conviction. In essence, Locke seems to attribute to S2 a kind of radical clause of conscience.

Leibniz mitigates this radical stand: “The argument ad ignorantiam is valid in cases of presumption where it is reasonable to hold to an opinion till the contrary is proved” ([1765], p. 576).
Presumption here has the meaning of burden of proof. The pretension of L1 may be excessive and misleading, but his argument nevertheless creates a preference in the field concerned, and in practice we can stick to it until something else has been proven.
This “for lack of anything better” reasoning seems to be the standard case in practical argumentation when a decision has to be made and a possibly urgent action has to be taken:

S1_1:     — Upon such and such basis, I propose 1) that we take such and such a disposition; 2) that we explore such and such a hypothesis. Now, the floor is yours

S2:        [Long silence]

S1_2:     — Nothing to say? Silence meaning consent,
1) In the absence of contradiction, my proposal is adopted.
2) In the absence of any other hypothesis, mine will serve as a working hypothesis.

It is difficult to object to S1_2’s conclusions. He or she does not claim that his proposition is the only viable one, nor that his hypothesis should be held to be true.

2. Ignorance and principle of the excluded middle

The argument from ignorance is also defined, without consideration of the quality of the argumentation, as an illegitimate application of the principle of the excluded middle:

P is true, since you are unable to prove that it is false.

The argument is not conclusive. If we consider that “not-P is not proven” is equivalent to “not-(not-P)” we conclude that P, by application of the principle of the excluded middle. But the two nots are not of the same nature: “not-P is not proved” does not mean “not-P is false”, which would be a confusion between what is true (alethic) and what is knowable (epistemic), S. Absurd.

3. Ignorance, burden of proof, precautionary principle

I am innocent, since you are incapable of proving that I am guilty.
You are guilty because you are incapable of proving your innocence.

Admitting that P is true, or acting “as if” it was true in the absence of proof of not-P is a decision that falls to the institution empowered to discuss and rule on such matter in the field concerned. In the judicial field, presumption of innocence places the burden of proof on the prosecution and gives the benefit of doubt to the defendant.

Precautionary principle
In the debate on the safety or toxicity of new products, a decision has to be made in a situation of insufficient knowledge. The presumption of safety would be:

Possibly the product has toxic effects, but this is not proven. So it has no toxic effects.

The precautionary principle is easiest to rebut when maximized:

Every new product is assumed toxic and will remain forbidden until its safety has been proved.

Under its common form, it simply reverses the burden of proof:

The precautionary principle (or precautionary approach) to risk management states that if an action or policy has a suspected risk of causing harm to the public, or to the environment, in the absence of scientific consensus (that the action or policy is not harmful), the burden of proof that it is not harmful falls on those taking an action that may or may not be a risk.
Wikipedia, Precautionary Principle

Situation: no scientific consensus on the inocuity of a given product
Decision: the burden of proof is upon those which use it

4. Argument from ignorance and argument from silence

S. Silence