\(C=2\pi{r}\)

Is your conclusion related to a premise by some multiple of \(\pi\)?

There's an increasing trend in apologetics, arising from a presuppositionalist stance, that horribly abuses a fairly well-established principle of logic, specifically a known fallacy, and I've seen some pretty poor approaches to addressing it. I'll start with an example. I'll note that the particular example I've chosen is one of the better approaches I've seen, and the atheist here - Adam Johnson - clearly grasps logic well and handles the situation admirably. However, he misses a trick.

The apologist - Dustin Segers - insists that reason is required to validate reason, and that this is circular. This would be a valid line of objection if it weren't for a couple of fatal problems.

The proper term for the fallacy cited by the apologist is petitio principii, or begging the question. 

It's important to note that some care is required here, because not all question-begging is circular reasoning, but all circular reasoning is question-begging.

So what do we mean by circular reasoning? We looked at this fallacy briefly in Deduction, Induction, Abduction and Fallacy, where we noted that:

This is one of the most common fallacies you're likely to come across. It's committed when the conclusion of an argument is contained within the premises. It's generally easy to spot but, occasionally, the conclusion is 'smuggled' into the premises in a form not easy to spot. It will usually be in the form of a premise whose truth relies on the truth of the conclusion (hence begging the question) or vice versa. Note that a question-begging argument is always valid and, if the premises are true, then it's also sound (it couldn't really be other in cases where the conclusion is contained in the premises because, if the premise is true, and contains the conclusion, the conclusion must be true). Gary Curtis gives the following example:

P1. Murder is morally wrong.
P2. All abortions are murders.
C. Therefore, abortion is morally wrong.

This is clearly a valid argument, and it doesn't appear that the conclusion is in the premises, so what's the problem?

It becomes clearer when you strip away the loaded term 'murder'. The first step is simply to remove the first premise, which lends nothing significant to the argument, and then to replace the word 'murders' with something more neutral. Then we're left with:

P. All abortions are wrongful killing.
C. Therefore, abortion is morally wrong.

In content, this argument is identical to the original, yet now the circularity is crystal clear.

So, if it's a valid argument, what's the problem?

The answer to this question should be reasonably obvious, but it's such a common fallacy that appealing to the fact that it's obvious is probably not going to get us anywhere. 

The problem lies in the fact that this is a deductive argument. Recall that, in a deductive argument, if the premises are true, and the reasoning valid, the conclusion is necessarily true. When I say 'necessarily', I'm using the full technical definition of necessity, in which the conclusion 'cannot fail to obtain'.

Of course, since the conclusion is what the argument sets out to prove, having the conclusion in the premises means that what we're saying is that if the conclusion is true, then the conclusion is true. Not only is this tautological, it means that we haven't actually proved anything, because it's tantamount to simply asserting blindly that the conclusion is true, which is itself a fallacy (ipse dixit - he said), as discussed in Onus Probandi, Assertionism and Peer-Review. In short, we simply can't say that the argument is sound, because the conclusion stands unsupported.

So, does this apply to science and reason? Let's start with science. We'll leave reason for the moment, not least because reason is at the root of it all. For the purpose of advancing the subject, we'll simply operate on the principle that reasoning works and circle back to it later (see what I did there?) for the coup de grâce.

The first place to look is at whether science is deductive. There are certainly deductive elements to the logic of science, but it isn't deductive in its entirety, and it's doubtful that those areas where it is are going to support a charge of circularity. To be rigorous, though, let's look at the process again, from the first cited post above:

1. Observe phenomenon.

2. Formulate hypothesis (abduction).
3. Compute the consequences of your hypothesis (deduction).
4. Compute a consequence that, if observed, will show your hypothesis to be incorrect (deduction).
5. Devise experiment (or observation) that will show one or other of the above.
6. Observe phenomenon:
7a. If 3 is observed, your hypothesis survives (induction).
7b. If 3 is not observed, your hypothesis is incorrect (deduction), and should be modified (at least) or discarded. 

7c. If 4 is observed, your hypothesis is incorrect (deduction), and should be modified (at least) or discarded.
8. Rinse and repeat.

Looking at the deductive portions of that, we can see that no charge of circularity can actually apply. Beginning at 3, the premise there is 'if my model is correct, we should see this'. Or, formally, \(P \Rightarrow Q\) (hypothesis \(P\) implies consequence \(Q\)). No circles to be had there.  We also have deduction in 4, but here the form is slightly different, namely 'if my hypothesis is correct, we definitely won't see this'. Formally, \(P \Rightarrow \lnot Q\) (hypothesis \(P\) implies not consequence \(Q\)). 

The next deductive portion is 7b, which is \(\lnot Q \therefore \lnot P\), while 7c is \(Q \therefore \lnot P\) (\(\therefore\) = therefore), which displays a suspicious absence of curvature. In content, we have the precise formulation of two of the most basic argument forms and, as rules of inference, they're pretty unassailable.  These are the forms of the modus tollens, or 'way that denies by denying'. Formally, and to show them completely, \(P \Rightarrow Q, \lnot Q \therefore \lnot P\) and \(P \Rightarrow \lnot Q, Q \therefore \lnot P\) respectively. In both cases, the conclusion is the negation of the premise, so it could hardly be said that the conclusion is contained in the premises. In each case, the logic is completely linear, and not even the remotest hint of an arc in sight. 

Note that, where a falsifying observation is made, this may not be the death of the hypothesis. Some falsifying observations will be so catastrophic to the hypothesis that it must be discarded completely, but we need to check it first. It may be that the computed consequences are incorrect, for reasons not yet elucidated. It may be that the falsifying observation highlights some limiting factor or effect that the researcher was unaware of while computing consequences. In all cases, she must go back to her calculations and check. 

All hypotheses, and indeed the broader theories they form part of, are apportioned degrees of confidence commensurate with Hume's famous dictum. They're used to generate predictions, and we accept that, when predictions are validated, our model survives, and we can apportion a bit more confidence. We state of such models that they are empirically adequate, and accept them tentatively pending future observations and theoretical developments that allow us to compute more consequences. Thus, because we're not employing our conclusions as deductive, and we're not asserting them as truth, the charge of circularity is fatally undermined.

Science is pragmatic and, as a result, self-correcting. Once the above is complete, the peer-review process begins. I deal with this in detail in Onus Probandi, Assertionism and Peer-Review - linked above - so I won't belabour it here. Suffice it to say for our purposes here that the entire system is geared toward showing that our model is wrong. Even for a theory that has withstood every bit of testing for centuries, we keep revisiting it, because a single observation is sufficient to fatally undermine a theory. We've covered this in various ways in the physics sections of this blog dealing with Newtonian mechanics and Relativity. Newton was having predictions validated to the eyeballs for more than 150 years, yet along came Einstein and showed that he was wrong. Not only are we actively working to break our theories when we first propose them, we're actively working to break them all the time. Breaking extant theories ishow Nobel Prizes are won.

As our understanding progresses, we gain more confidence in it, which makes our reliance on it as empirically adequate and unfalsified as any theory or hypothesis. In other words, our reliance on science is itself inductive, and each new triumph of prediction or explanation increases our confidence. We go from evidence, to hypothesis to computed consequences, to tentative conclusion, the conclusion in this case being:

Image: XKCD 

Incidentally, it's worth tracking the story of that graph, because it's one of the great triumphs of modern physics. There's a wonderful book by Marcus Chown called Afterglow of Creation, in which the above logic takes centre-stage, and the pragmatic nature of science is vindicated in absolutely spectacular fashion. I can't recommend this beautifully-written book highly enough.

So I've conclusively demonstrated that our reliance on science is anything but circular, but what about reason?

Well, here's the thing. The entirety of what's been said above can also be applied to reason, because the reasoning employed in science is exactly the same reason that we apply to reason itself. Ultimately, reason is pragmatic. We utilise all of the same principles in reason as we do with science, The logic is identical. We rely on reason because, once again, it works, bitches!

There's a famous old saw attributed to Heraclitus of Ephesus:

"No man ever steps in the same river twice, for it's not the same river and he's not the same man."

Circular? Yeah, right.

Thanks to Sendraks and Atyhans for their input in attaining clarity. Original video can be found on Eddie Becker's Channel.