Further Notes on Ampliative and Non-ampliative Inference

In a number of my previous posts I have drawn a distinction between ampliative and non-ampliative inferences. My purpose in doing so has been as part of a larger project to clarify, if only for myself, some of the details of the development of doctrine in the Church and how that development applies to such dogmata as that of the Trinity or the Filioque.

More recently, because of some postings by my friend Dr. Mike Liccione over at Sacramentum Vitae, there has been much combox activity from both Orthodox and Anglican readers regarding the precise nature of this claim that doctrine develops. This is a clarification of what I think is meant by the term.

Consider the following set of propositions.

1. If Bob went to the show, then Mary went to the show.
2. If Sally went shopping then Sarah went swimming.
3. Either Bob went to the show, or Sally went shopping.

I apologize in advance for the banality of the sample propositions here, but the semantic content is not as important as the logical structure at this point.

Suppose, for the sake of argument, that propositions (1)-(3) are the beliefs of a certain group of people, call them the Bobheads. If someone were to ask us, "What are the principle beliefs of the Bobheads?", we could enumerate propositions (1)-(3). Could we, however, say that the Bobheads also believe this proposition:

4. Either Mary went to the show or Sally went swimming.

It might be tempting for someone to say: "No, you cannot impute that belief to the Bobheads, because it is not one of the propositions that they explicitly endorse in their list of beliefs. However, proposition (4) is a necessary consequence of beliefs (1)-(3). Logically, we can say that (4) is entailed by (1)-(3) by virtue of a rule of inference called "constructive dilemma": if propositions (1)-(3) are all of them true, then propositions (4) is necessarily true as well, and not just accidentally so, but because propositions (1)-(3) are true.

In this particular case it is rather easy to see the connection between propositions (1)-(3) and proposition (4). In other cases, however, the logical connection between a set of propositions and its entailments may not be so clear. The important point to note, however, is that proposition (4), although it says something that may appear to be a little different from what any of propositions (1)-(3) say individually, it does not, in fact, say anything the least bit different from what propositions (1)-(3) say collectively, that is, the information that is present in proposition (4) is not different from the information contained in propositions (1)-(3). This is precisely what guarantees the truth of the conclusion of a deductively sound argument. When the argument form is inductive rather than deductive, the information in the conclusion is actually new, and there is no guarantee that it is true even if all of the premises are true.

Inductions, then, are ampliative in the sense that they claim something above and beyond what the premises claim. Deductions, by contrast, are non-ampliative, because their conclusions do not state anything different from what the premises collectively tell us. Doctrine develops only deductively, not inductively hence, doctrine develops only in a non-ampliative manner.

In the case of the doctrine of the Trinity, for example, nobody who has actually read the New Testament with a critical and intelligent eye will claim that the doctrine of the Trinity is explicitly stated there in the way that the Bobheads' beliefs are explicitly stated in propositions (1)-(3). Rather, the doctrine, if it is in there at all, must be extracted by means of logical entailments of the sort that proposition (4) represents. Some of these entailments will be obvious, others less so. It is clear to everybody that God is One in Three; it is less clear that the Holy Spirit proceeds from the Son in the same way that he proceeds from the Father. Or rather, the logical entailment that necessitates the truth of that claim is not obvious to everybody.

An important worry that must be addressed in this connection is the following. What, precisely, is involved in saying that "I believe propostion p"? Must I be consciously aware of the semantic content of p in order to say that I believe p? For example, suppose you were to say to one of the Bobheads, "Oh, so you must think that Mary went to the show or Sarah went swimming." The Bobhead in question may very well say "No, I don't think either of those things happened." You might then say "But come on, you said that you do believe (1)-(3), so how could you not believe that either Mary went to the show or Sarah went swimming?" You might well say that. But the Bobhead might well look at you and say "I don't believe it because I only believe (1)-(3). Those are my beliefs. I don't have any others." If you are tempted to argue with the Bobhead, pointing out that (4) follows from (1)-(3), you might find yourself talking to a brick wall. If you have any doubts about whether people can refuse to believe things that they are logically committed to, try teaching a college student every now and then. Or really anybody: folks are remarkably irrational when it comes to acknowledging that they are logically committed to things that they may not see themselves as logically committed to.

And in fact it's not that obvious that they are being irrational anyway. I know what even numbers are, and I know what odd numbers are, and I know what prime numbers are. Do I also know the truth of the proposition "Every even number is the sum of two primes?" No, I don't, even though the truth or falsity of it is entailed by things that I do know--so how could I not know it? I just don't, because I have never bothered to trace out the proof (or refutation) of it. In fact nobody has, so nobody knows whether it's true or false, even though there are some great mathematicians out there who have worked on it, and presumably they have even more true beliefs about numbers than I do.

So it's quite possible for folks to be committed to believing things that they do not, in fact, consciously believe and that, indeed, they may consciously deny believing, and yet they are not necessarily being irrational. They merely need to find the proof or refutation and see the logic of the situation. That is what real ecumenical dialectic ought to be about.

If it really is true that, say, the Filioque as the West understands it follows from the truth of what both East and West believe, as St. Anselm holds, then eventually there will be agreement on the truth of that doctrine, and it will not be a new truth, but a discovered truth, not discovered in an ampliative sense, but discovered in a logical sense, in the sense that we now see how it follows of necessity from what we all agree to be true. Possibly we will find that it does not follow, or that it is contradicted. That would be bad from a Western point of view, and I don't think it possible, but it is certainly possible in other kinds of cases, and both sides need to be open to such possibilities.