# Can the AC-DC argument against infinitism be defused?

Infinitism is the epistemic theory that claims that justification is only achieved by an infinite chain of non-repeating reasons.

At first, this feels like the "troll" theory of epistemic justification. But perhaps it was unfairly dismissed — let's remember that only in the late 19th century, actual infinity (not just potential infinity) was taken seriously and accepted.

Now there's the AC-DC argument against infinitism which seems very powerful to me (perhaps even more than the finite mind objection):

For any belief P we can construct an Affirmation Chain (AC), justifying P:

Q & (Q → P)

R & (R → (Q & (Q → P)))

S & (S → (R & (R → (Q & (Q → P)))))

and a Denial Chain (DC), justifying the negation of P:

Q & (Q → ~P)

R & (R → (Q & (Q → ~P)))

S & (S → (R & (R → (Q & (Q → ~P)))))

IMHO, the problem of such chains is the emptiness of the inferences. In this case, they are even logical / analytical. Still, we have to wonder why infinitism suffers from this, since foundationalism does not.

Also, it's not so difficult to construct non-logical examples. E.g. the "proof" that 2 is an odd number, where we use the simple theorem that if an integer n is even/odd, then n - 2 will be, too. Here's the wrong denial chain:

2 is an odd number

2 is an odd number, because 4 is odd and 4 - 2 = 2

4 is an odd number, because 6 is odd and 6 - 2 = 4

And the same manner of "proof" can be used for the (correct) affirmation chain.

But is the AC-DC objection really more powerful than the utterly trivial objections against the other theories, that your foundational belief might be wrong, or that you constructed a set of cohering false reasons? Those theories also need some special "ingredient".

It seems, infinitism puts the focus on the inferences which have to do more than just procrastinating to give any sort of enlightening inference by pointing at the next reason, and the next, and the next. Some evil-minded creativity is needed to even construct such empty, infinite chains — which don't fool anyone, since they are kind of obvious in their utter emptiness.

Still, I don't feel convinced. Because we can't give a concrete answer about what actually went wrong, i.e. where the error occurred — contrary to foundationalism or coherentism.

So can we really defuse the AC-DC objection? "Emptiness of inferences" is way too vague, at least compared to the explanations of failure for foundationalism and coherentism.

• "the simple theorem that if a natural number n is even/odd, then n - 2 will be" That is not a theorem. 2 is a counterexample proving that it is not a theorem. Jan 12 at 18:50
• @DavidGudeman why is zero not an even number? Zero is dividable by two without remainder. Anyway, of course at some point we reach negative integers, which aren't natural numbers. I'm really not here for mathematical precision ... it's kind of ... obvious. Jan 12 at 20:55
• It doesn't matter whether 0 is an even number or not. The point is that the theorem you mentioned is not a theorem, and so the issue that you can derive a false result from it is not interesting. Jan 12 at 23:36
• @DavidGudeman Why don't you just explain why it is not a theorem? If n = 2 is even, then n - 2 = 0, which is an even number. So 2 is not a counterexample. The proof for the theorem is: If n is even, then there is an integer m for which we have n = 2 · m. Then n - 2 = 2 · m - 2 = 2 · (m - 1), and m - 1 is an integer, so n - 2 is even. qed. Jan 13 at 14:38
• If 2 is not a counter-example then 0 is. Jan 13 at 15:15

Infinitism is the epistemic theory that claims that justification is only achieved by an infinite chain of non-repeating reasons.

The critical aspect of infinitism, namely that it involves an infinite number of justified beliefs is discussed in the SEP:

Infinitism is a view that should be seriously considered, particularly once one realizes that one not only can but arguably does have an infinite number of justified beliefs (e.g., that 2 is greater than 1, that 3 is greater that 1, and so on.) -- Ali Hasan, Foundationalist Theories of Epistemic Justification (SEP, 2022)

Words are really cheap. The idea that one has an infinite number of justified beliefs is patently false. The human brain's cognitive capabilities are finite and we could not possibly have any infinite number of anything, beliefs or otherwise.

This is rather painful to watch. The author takes the pain to explain the various theories of knowledge, in particular the idea that we have to justify our beliefs by logical inference from other beliefs we already have. Thus, believing that 3 is greater than 1 requires to believe for example that 2 is greater than 1 and 3 greater than 2, and that x greater than y and y greater than z implies x greater than z. This is already quite something, just to justify that 3 is greater than 1! Yet, this is not even a complete justification. Mathematicians have invented an axiomatic theory pretending to explain all this but their theory is clearly not what the human mind really does. For one thing, nobody outside academia ever bother to justify anything. People just decide intuitively that 3 is greater than 1. The fact that we think we could contrive a rather complicated justification is neither here nor there. Most of our beliefs simply don't need justification, and, the beauty of it is that we believe that most of our beliefs which really don't need justification could nonetheless at least in principle be justified, except that in practice, no one ever actually ever did that. Too bloody complicated. We believe what we believe and this will have to be good enough.

Now, imagine an infinite number of justified beliefs... Clearly, this is a non-starter. The brain is a finite thing, and coding information requires energy, time and space. In fact, this is why we need logic. Our brain uses logic to swap time for space. Our brain prefers to spend time on a small number of inferences as circumstances dictate in order to avoid having to store all the information that might overwise one day be needed. If we could have an infinite number of justified beliefs, we wouldn't even need logic.

And I don't believe for a moment that anyone really believes that they have an infinite number of justified beliefs. Sure, we could in principle justify that each natural number is greater than other natural numbers, but this is only in principle, we don't actually do it. Whatever category of things we might be able to think of, there is only a finite number of categories, and a finite number of things that we could identify as belonging to this category.

The AC-DC argument suffers from the same problem, for not only is there no actual infinity of justifications available to us, but there is a practical limit to their number which makes the AC-DC argument an empty claim.

For any belief P we can construct an Affirmation Chain (AC), justifying P

Can we? If we could, we would have already done it. Science is the extremely painstaking business of justifying some of our most important beliefs. The problem of science is that the cost of justification goes exponentially up as it digs deeper into the fabric of reality, and this in terms of both time, energy and space. There never was such a large number of active scientists as there is today, and they keep producing better science, but the cost of it is already putting a limit on what even the richest countries in the world are prepared to pay for. Science itself has become a necessary business precisely because the capacity of each human on their own is so limited, but it is only a matter of time before modern science itself reaches its own limits, although in practice science will just go ever more slowly, without necessarily "ever" stopping.

The fundamental mistake of these "theories" is to assume that logic is some sort of Platonic truth applying outside any physical constraints. The reality is that logic is a cognitive capacity, not any formal system, and it works like any cognitive capacity, that is, within rather narrow limits. Further, humans don't use logic to justify their own beliefs. It is good enough to believe something. And then when we use not logic, but logical reasoning, it is to convince other people, and then only when they happen to not share some belief with us. And we do it on the cheap, because if we were really trying, we would spend so much time and energy doing it that humanity would disappear at short notice. Scientists can afford to do what they do only because governments, or rather other people, pay their living costs, a principle which clearly has its own limits.