The Logic of the Infinite Regress

On the one hand, it’s essential to understand logical errors. On the other, it’s maddening. The clearer you think, the clearer you see the world, the clearer you see the errors in other people’s reasoning – which tend to be numerous, repeated, and sometimes aggressive. One logical error which is commonly misunderstood, and often outright defended, is the infinite regress.

The idea is simple to illustrate. Take the following conversation:

Bob: “Proposition Z is true.”

Joe:  “What’s your justification for proposition Z?”

Bob: “Proposition Y.”

Joe: “And what’s your justification for proposition Y?”

Bob: “Proposition X.”

Joe: “And what’s your justification for proposition X?”

Bob: “Proposition W…”

And so on, ad infinitum.  For every proposition, Joe can respond, “And what’s your justification for that?”

Thus, We Conclude

If this chain of reasoning has no end, then we’ve no reason to believe proposition Z or any other proposition in the chain is true. That might not seem intuitive, so I’ll explain.

Examine conclusion Z.  We’ve only one reason to believe it’s true: if, and only if, its justifying premise – proposition Y – is true. If proposition Y is false, then we’ve no reason to believe conclusion Z is true.

The same is true for proposition Y. We’ve only one reason to believe it’s true: if, and only if, its justifying premise – proposition X – is true. If proposition X is false, we’ve no reason to believe proposition Y is true.

And of course, the same is true of proposition X.

This means, each proposition in the chain – without exception – is contingent on its preceding premises. Each proposition is like an empty vessel, dependent on the truth-value of the premise before it. If there’s a falsehood anywhere in the chain, it poisons every conclusion which follows. If each link in the chain is true, then we can trust the conclusion (assuming the reasoning is sound).

But when we interject infinity into the chain, we’re presented with a problem: we’re never given a reason to believe any proposition in the first place – i.e. there is no truth to transmit.

The error is similar to the Liar’s Paradox; you never end up with a concrete proposition to evaluate as true or false.

By definition of what we mean by “infinity”, no proposition contains a truth value by itself. It’s a chain of empty vessels; regardless of how many there are, you’ll logically never end up with justification for a single proposition.

It’s like adding up an infinite amount of 0’s to try to get 1. It will never work, no matter how many zeros you add together.

Every proposition is contingent – contingent on other contingencies.  If we were to ask, “Ultimately, what is conclusion Z justified by?”, the only logically consistent answer is to say, “Nothing”. If you ever end up with a non-contingently true premise, you’re not dealing with an infinite regress.

Thus, by logical necessity, any argument which falls into an infinite regress is foundationless. There’s no reason to believe any proposition in an infinite chain – because there’s no real justification to be found, by definition.

Big Implications

Many skeptical philosophers have profoundly concluded: therefore, all knowledge is without foundation. No non-contingently true premise exists, therefore we cannot know anything at all.

Of course, if you follow my work, you know I strongly disagree. While it’s true that an infinite regress is a logical error, not all propositions are contingent truths. Chains of reasoning can be grounded in a bedrock foundation: logical necessity. You can start from certain premises with known truth values. Self-evident truths break any regression of “Why?” questions.

The logic of infinity is not only related to epistemology. Profound conclusions (and errors) about infinity permeate mathematics, as well. Perhaps most historically significant, many philosophers have argued for the existence of an “uncaused cause” of the universe, in order to avoid an infinite regress of causal events. Many thinkers have identified this with “God”. This so-called “cosmological argument” is so significant, that I will devote an entire article to it.

So, you can be sure of inaccurate reasoning whenever you see somebody at peace with an infinite regress. Humans are notoriously lazy when it comes to thinking about infinity, so most people think, “Even with an infinite regress, at some point, my conclusions are justified.” When, in reality, the opposite is the case. Logically speaking, at no point does an infinite regress justify any proposition.