To begin, I'm not sure that we are really talking about precisely the same situation. Consider the following, from Mike's post:
In the lingo of logicians, "inductive" inference is any type of inference that is valid, in principle and for certain purposes, but not strictly deductive. The most common sort of induction is inferring that the future will, in this or that respect, be like the past.Technically, the term "validity" refers to a property that applies strictly to deductions only, so it seems to me one of two things is going on here. Either Mike is using the word "valid" in a non-technical sense to mean something along the lines of "rationally warranted inference" or else he has something rather unorthodox in mind when he refers to an inductive inference. Since he goes on immediately to give a perfectly orthodox example of an inductive inference, I must assume the former.
So far, so good: after many years of teaching logic I know only too well that most people, when they say that an inference is "valid", do not intend to say that it is valid in the technical sense. Most folks are happy just to say that they think they can see where a particular inference comes from, and that is generally what they mean by the word "valid". Technically, every inductive inference is invalid, since it is always possible for the truth of an inductive inference to be false even if all of the premises are true. For this reason we do not characterize inductive inferences as either valid or invalid; they are characterized rather as being either strong or weak.
This is an important point, because nobody is claiming that inductive inferences should not be persuasive--a well-constructed induction should have certain properties that persuade rationally disposed hearers to believe its conclusion to be true with a certain degree of confidence less than 100% but greater than, let's say, 50%. But there is a huge difference between persuading somebody that a particular claim is true and proving that it must be true of necessity. Only a deductive inference can accomplish the latter, and only when it is sound (that is, it is valid in form and all of its premises are indisputably true).
My claim in Further Notes was that doctrine only develops in accordance with deductive principles. I confess that at least part of the motivation behind my claim was political in an ecumenical sense: I want to set at ease the hearts and minds of those Orthodox who worry that the principle of the development of doctrine warrants introducing radically new (and possibly heretical) belief-statements into the corpus of beliefs that must be held de fide. I think that this is a "valid" worry (non-technical use of "valid" here), and it is one that I share. The difficulty with any and all inductive inferences is that they are subject to (often massive) underdetermination, that is, the evidence can never establish the truth of any particular inference to the exclusion of all competing, non-consistent inferences. This is not a situation in which we want to find ourselves when trying to discover what must be believed de fide.
Certain kinds of scientific realists have suggested that this worry is overblown. They characterize certain kinds of inductive inferences as having far greater warrant than others. In order to give a certain cache of respectability to this claim they have actually come up with a special name for this kind of inductive inference: abduction. An abductive inference is one that is supposedly more likely to be true than its competitors, even though the same evidence is available to all candidate inferences. It is sometimes called an "inference to the best explanation" on the grounds that it posits an explanation that is inherently more plausible than its competitors. For example, suppose I find little teeth-marks in the chunk of cheese on my kitchen counter, and I hear little scratching sounds in the walls, and I find little tiny turds all over the floor and counter-tops. Now imagine two explanations. According to explanation (A), there is a mouse in my house. According to explanation (B), my evil little brother is trying to freak me out and drive me batty by setting things up in my kitchen to make it look as though I have a mouse in there when he knows that I have a very important dinner party coming up. Both explanations are consistent with all the known (and knowable) evidence (assume that my little brother really is such a person as to do such a thing), yet (A) seems, somehow, more plausible than (B), if only because it seems less ad hoc. But an abduction is just an induction by another name, and it suffers from all the same problems that plague induction generally. I might commit myself to (A) only to discover that it was, indeed, my little brother all along, and nothing about the evidence itself favored (A) over (B)--the only thing that makes (A) more likely than (B) is a set of theoretical presuppositions that I bring to bear on the inference-drawing process itself.
Now, Mike suggests that there are, nevertheless, gradations of some kind among inductive inferences, and in particular he wants to claim that an inferential pattern that he thinks he has found in the Scriptures rises above the merely inductive to something like the abductive. He gives a rather intriguing example:
Now I consider it fairly obvious that some, perhaps even much, DD is ampliative and thus "inductive" in logicians' lingo. To take my favorite example, that the Son is homoousios (of the same substance) as the Father does not follow by strict deduction from the testimony of Scripture, the Apostles' Creed, and what the various liturgical rites of the early Church all had in common. If it did, then all it would have taken to refute the Arians decisively, once for all, would have been a logical proof of the sort that had long before been provided for, say, the Pythagorean Theorem in geometry.I'm particularly concerned about Mike's claim at the end here, that the logic of deduction is always going to be something that just pops out and is obvious to everybody. Now Mike himself noted, in a post at his own blog, that he accepts Saint Anselm's argument in favor of the Filioque, and perhaps he means something non-standard when he says he buys that argument but when I say things like that I mean that I think that the argument works. Anselm's argument is a deductive argument, and it is laid out with what can only be characterized as anal-retentive precision and care, and yet it failed to persuade the Greeks. I do not think that it is the case that a deductive argument is always going to do the trick when it comes to "refuting" any particular inference, if all we mean by "refuting" is getting folks to acquiesce in our take on things. If he means something more technical by "refute"--if he means that the Arians were proven wrong, then of course they were refuted. They just didn't think that they were. In short, I think that the homoousios doctrine does follow by strict deduction. This is not to say that every premise needed for that deduction is made explicit in Scripture; some premises are themselves intermediary conclusions of other deductions. But the inference itself needs to be deductive or else there is no rationally compelling reason to believe it.
The notion of compulsion here is extremely important--it is not just a rhetorical nicety to stick in a word like that. Inductive inferences have a certain rational warrant to them--that is, if they are strong it is not irrational to accept them--but precisely because of underdetermination there is no compulsion to believe them--we may always question any inductive inference and in so doing we still act rationally. It would be irrational, by contrast, to question a sound deductive inference.
This brings me to a point that Mike makes a little later in his post. In fishing around for a description of the pattern of inference he is talking about he writes:
I can't think of a name offhand, but I believe I can see the pattern in the unfolding of divine revelation itself. Consider how Matthew 1:23 cites Isaiah 7:14 to support the claim that Jesus was born of a virgin. Matthew was relying on the Septuagint translation of the Hebrew Scriptures into Greek, which uses the term parthenos, meaning "virgin," to translate Isaiah's almah, meaning "young woman." Why that translation? After all, not all virgins are young women and not all young women are virgins. Perhaps the "seventy" Jewish scholars in Alexandria who produced the LXX believed that the Messiah would be born of a literal virgin; but then, perhaps not. We really don't know. They may simply have chosen parthenos as a decorous synonym for 'a young woman' with the implication that the Messiah would be her first-born. At any rate, we have no evidence that first-century Jews assumed the Messiah would be born of a literal virgin. There doesn't appear to have been any consensus among Jews about how to construe Isaiah 7:14 on this particular point. Yet Matthew, or at least the early Church that received his Gospel as canonical, seems serenely confident that it prophesied that Jesus the Messiah was born of a literal virgin.The difficulty here is that this is not an inference at all, but an interpretation. While it is true that interpretation of data is a necessary condition on any inductive inference, it is of course also true that every deductive inference requires interpretation of data. So the fact that this pattern is to be found in the "unfolding of divine revelation" is insufficient to show that the pattern of inference involved in the development of doctrine is not deductive.
It's also worth pointing out that, in this particular case, anyway, we're dealing with scriptural claims, albeit claims separated in time. From our perspective, though, the claims of Scripture themselves do not develop, but rather our own understanding of their meaning does. To take just one of the more familiar instances of this sort of thing, consider the Church's teaching on usury. This is a favorite canard of the cafeteria catholic crowd that seems to take a perverse pleasure in saying that doctrine develops in a way that will ultimately result in new teachings on women priests, gay marriage, and a whole slew of other complaints-du-jour. It is precisely this sort of crap that a real principle of the development of doctrine must be well-constructed enough to avoid. Usury has always been condemned by the Church, but what the Church is willing or unwilling to count as an instance of usury is a prudential judgment that is contingent on certain economic and social facts. These facts, obviously, change over time, as economic and social conditions change. But the morality of usury itself never changes because it is not grounded in contingent judgments but necessary ones. (The same is true of torture, by the way, as I've pointed out in many posts. Check the archives if you're interested.)
How do we know that the Church is right about what counts as usury these days? We don't. Nor do we know for certain that capital punishment will never again be necessary for the defense of the common good. We know with certainty that usury is wrong, but we cannot know with certainty whether a particular rate of exchange is usurious in every possible case. But by becoming Christians we do place ourselves under the authority of others: we trust in the Church's authority to teach us in matters of faith and morals, even on those occasions (not rare, but not ubiquitous) when those teachings are grounded in contingent or prudential judgments. It may be in this sense that there is ampliative development of doctrine: once the deductions have been carried out, we require some authoritative source of interpretation. I said above that a sound argument cannot rationally be disputed, and a sound argument is one that is both deductively valid and its premises are all indisputably true. When was the last time you saw a premise that was indisputably true? Every premise is disputable in some sense. So in order for any teaching to be held with some degree of confidence, we must place our trust in some source of interpretive authority, and that source of interpretive authority is the Ordinary Magisterium.