Definitions

In the previous post, I offered a criticism of defining logical consequence by means of proofs. A more precise way to put my criticism would be:

1. Logical consequence is equally well defined by (i) tree-proofs or by (ii) Fitch-proofs.

2. If (1), then logical consequence is either correctly defined by (i) and correctly defined by (ii) or it is not correctly defined by either.

3. If logical consequence is correctly defined by one of (i) and (ii), it is not correctly defined by the other.

4. Logical consequence is not both correctly defined by (i) and and correctly defined by (ii). (By 3)

5. Logical consequence is neither correctly defined by (i) nor by (ii). (By 1, 2, and 4)

When writing the post I had a disquiet about the argument, which I think amounts to a worry that there are parallel arguments that are bad. Consider the parallel argument against the standard definition of a bachelor:

6. A bachelor is equally well defined as (iii) an unmarried individual that is a man or as (iv) a man that is unmarried.

7. If (6), then a bachelor is either correctly defined by (iii) and correctly defined by (iv) or it is not correctly defined by either.

8. If logical consequence is correctly defined by one of (iii) and (iv), it is not correctly defined by the other.

9. A bachelor is not both correctly defined by (iii) and correctly defined by (iv). (By 9)

10. A bachelor is neither correctly defined by (iii) nor by (iv). (By 6, 7, and 10)

Whatever the problems of the standard definition of a bachelor (is a pope or a widower a bachelor?), this argument is not a problem. Premise (9) is false: there is no problem with saying that both (iii) and (iv) are good definitions, given that they are equivalent as definitions.

But now can’t the inferentialist say the same thing about premise (3) of my original argument?

No. Here’s why. That ψ has a tree-proof from ϕ is a different fact from the fact that ψ has a Fitch-proof from ϕ. It’s a different fact because it depends on the existence of a different entity—a tree-proof versus a Fitch-proof. We can put the point here in terms of grounding or truth-making: the grounds of one involve one entity and the grounds of the other involve a different entity. On the other hand, that Bob is an unmarried individual who is a bachelor and that Bob is a bachelor who is unmarried are the same fact, and have the same grounds: Bob’s being unmarried and Bob’s being a man.

Suppose one polytheist believes in two necessarily existing and essentially omniscient gods, A and B, and defines truth as what A believes, while her coreligionist defines truth as what B believes. The two thinkers genuinely disagree as to what truth is, since for the first thinker the grounds of a proposition’s being true are beliefs by A while for the second the grounds are beliefs by B. That necessarily each definition picks out the same truth facts does not save the definition. A good definition has to be hyperintensionally correct.

Inferring an "is" from an "ought"

You tell me that you saw a beautiful sunset last night. I conclude that you saw a beautiful sunset last night. You are talking about Mother Teresa. I conclude that you won't say that she was a sneaky politician. You promise to bake a pie for the party tomorrow. I conclude that you will bake a pie for the party tomorrow or you will have a good reason for not doing so. I tell a graduate student to read a page of Kant for next class. I conclude that she will read a page of Kant for next class or will have a good reason for not doing so.

All of these are inferences of an "is" from an "ought". You ought to refrain from telling me you saw a beautiful sunset last night, unless of course you did see one. You ought not say that Mother Teresa was a sneaky politician, as she was not. You ought not fail to bake the promised cake, unless you have good reason. The student ought not fail to read the Kant, unless she has good reason.

All of these are of a piece. We have prima facie reason to conclude from the fact that something ought to be so that it is so. In particular, belief on testimony is a special case of the is-from-ought inference.

In a fallen world, all of these inferences are highly defeasible. But defeasible or not, they carry weight. And there is a virtue—both moral and intellectual—that is exercised in giving these inferences their due weight. We might call this virtue (natural) faith or appropriate trust. We also use the term "charity" to cover many of the cases of the exercise of this virtue: To interpret others' actions in such ways as make them not be counterinstances to the is-from-ought inference is to charitably interpret them, and we have defeasible reason to do so.

The inference may generalize outside the sphere of human behavior. A sheep ought to have four legs. Sally is a sheep. So (defeasibly) Sally has four legs.

I used to think that testimony was epistemically irreducible. I am now inclined to think it is reducible to the is-from-ought inference. Seeing it as of a piece with other is-from-ought inferences is helpful in handling testimonial-like evidence that is not quite testimony. For instance, hints are not testimony strictly speaking, but an inference from a hint is relevantly like an inference from testimony. We can say that an inference from a hint is a case of an is-from-ought inference, but a weaker one because the "ought" in the case of a hint is ceteris paribus weaker than the "ought" in the case of assertion. Likewise, inference from an endorsement of a person to the person's worthiness of the endorsement is like inference from testimony, but endorsement of a person is not the same as testimony (I can testify that a person is wonderful without endorsing the person, and I can endorse a person without any further testimony). Again, inference from endorsement is a special case of is-from-ought: one ought not endorse those who are not worthy of endorsement.

If is-from-ought is a good form of inference, the contraposition may-from-is will also be a good form of inference. If someone is doing something, we have reason to think she is permitted to do it. Of course, there are many, many defeaters.

It is an interesting question whether the is-from-ought inference is at all plausible apart from a view like theism or Plato's Platonism on which the world is ultimately explanatorily governed by values. There may be an argument for theism (or Plato's Platonism!) here.

The PSR and inference to best explanation

Sam Cole, one of the students in my upper level metaphysics class, wrote an interesting paper (I am writing this with his permission) where he argued that if we do not accept the Principle of Sufficient Reason (PSR), then the following question will be unanswerable:

1. Under what circumstances should we accept a given explanatory hypothesis instead of the hypothesis that the phenomenon in question simply has no explanation?

I think this is a really neat question. We have some idea of the sorts of criteria we employ in choosing between alternate explanatory hypotheses: simplicity, prior probability (perhaps I repeat myself), etc. But if we do not accept the PSR, then the no-explanation hypothesis is going to be, presumably, always available. On what grounds do we judge between our best explanatory hypothesis and the no-explanation hypothesis?

It is tempting to say: If the best explanatory hypothesis is pretty good, then we go for it. But the evaluation of the quality of hypotheses seems to be innately comparative. So this "pretty good" does not seem like it should be absolute. But if it is relative, then what is it relative to? If it is relative to other explanatory hypotheses, then its being "pretty good" seems irrelevant when comparing it against the no-explanation hypothesis. The hypothesis that JFK was shot by a bunch of gorillas escaped from the zoo is pretty good as compared to the hypothesis that JFK was killed by a rifle-toting clam, but that is irrelevant when we compare the gorilla hypothesis to the Oswald hypothesis. So what we need to know is whether the explanatory hypothesis is "pretty good" as compared to the no-explanation hypothesis. But we have no criteria for that sort of comparison!

Another tempting suggestion is this: Whenever any narrowly logically coherent explanation has been offered (asking for more than that may run into Kripkean problems), we should reject the no-explanation hypothesis. This is a more promising answer to (1). Note, however, that an opponent of the PSR who takes this route cannot oppose the use of the PSR in the Cosmological Argument. For in the context of the Cosmological Argument, the PSR is employed to claim the existence of explanations for phenomena for which narrowly logically coherent explanations—namely, theistic ones—have indeed been offered.

Duct tape and naturalism

A parable

Fred buys a used car in terrible shape. Off the top of his head, he can see a dozen problems with it: the window is cracked, the engine doesn't start, the window washer liquid line is leaking, the steering is gone, etc. He knows little about how cars work, but some things are obvious. He takes a roll of duct tape, and immediately fixes three problems: puts lots of tape on the window to keep the glass in place, patches the window washer liquid line, etc. With a bit more thought, he can find three more of the twelve problems that, with some ingenuity, he can fix with duct tape. While doing this, he discovers two new problems with the car. He then struggles and struggles, and with a great deal of cleverness manages to fix a steering shaft broken in half with just duct tape. He is really proud of his solution--it involved stretching the duct tape very, very thin, then put down many thin layers of duct tape over the break, and finally melt the layers together, with that the resulting shaft having almost the strength of unbroken steel.

So now Fred knows of 14 problems with his car, of which he's fixed seven with duct tape. Question: How much reason does Fred have to think that he can fix all the remaining problem--the seven he knows of plus whatever ones he would discover while fixing those--with duct tape as his only material? Answer: Very little (particularly if he notices that the reason the engine doesn't start is because there is no spark plug). One might try an inductive argument: all of these particular seven problems were solvable with duct tape, and hence so are the others. But this argument fails due to an egregious bias in sampling: the seven problems just are the problems that Fred found himself capable of solving with duct tape. They were selected for their solvability. The other problems are ones that Fred couldn't see how to solve with duct tape.

Likewise, if we ask whether Fred has much reason to think that of the remaining solvable car problems, the solutions all involve only duct tape, the answer is negative, for exactly the same sampling bias problem. We may, however, have some inductive data that of the remaining car problems solvable by Fred, the solutions will all be based on duct tape, since it seems that Fred doesn't know any other way to solve problems.

Application

Consider this argument for naturalism: We've been able to solve many, many explanatory problems naturalistically. Hence, the many remaining explanatory problems which we do not at this point know the answer to are also solvable naturalistically if they are solvable at all, and naturalism is true.

The same reason that the argument for Fred's ability to solve the other problems with duct tape was bad shows that this is a bad argument. The set of problems that we've solved naturalistically is not a random sampling of the explanatory problems. Rather, it just is the set of problems that we've solved naturalistically.

One might try to strengthen the argument for naturalism by adding that no explanatory problems we know of have non-naturalistic solutions, and concluding, inductively, that all the explanatory problems that have solutions have naturalistic solutions, which could be enough to make naturalism plausible. But this argument becomes question-begging against typical non-naturalists who claim that they have non-naturalistic solutions to a number of vexing problems (intentionality, free will, normativity, the origin of life, the origin of consciousness, the origin of space-time, the origin of mass-energy, the origin of contingent things, etc.) Moreover, even if all of those can be argued not to work, it could well be the case that the problems that have non-naturalistic solutions are much harder to solve, perhaps even are not solvable by humans; nor is this some ad hoc posit, but simply comes from the that fact non-naturalistic things are not amenable to empirical study. Thus even if it were true that all the explanatory problems that we have solved have naturalistic solutions, it would not give us much reason to believe that all explanatory questions that have answers--including those that have answers that are beyond our capabilities--have naturalistic answers.

The deposit of faith

Consider the following objection to the Catholic faith (this is based on something I got by email): Catholicism includes a large number of detailed and substantive doctrines that do not seem to be derivable from God's revelation as completed by around the time of death of the Apostles, even though the Catholic Church herself claims that revelation was completed by around the time of death of the Apostles.

Consider, after all, something like the doctrine that Mary was free of original sin from the first moment of her conception. This is a detailed and substantive doctrine that seems to go far beyond the information given in Scripture and what we know about the faith of the first century Church from non-Scriptural sources. The objection is an incredulous stare at the possibility that such doctrines could be derived from revelation as completed by around the time of death of the Apostles. But:

1. Twenty simple axioms of Euclidean geometry generate an infinity of detailed and substantive theorems. These theorems are such that there is no prima facie way to see that they would follow from the axioms. It can take centuries and centuries for humankind to discover that they can be derived. It should, thus, be no surprise at all that we can derive from a set S of propositions new propositions that are details and substantive, and that seem to go far beyond S. This is particularly true when S is not a list of twenty axioms, but includes about 27,570 verses of the Old Testament, about 7956 verses of the New Testament, as well as decades of Apostolic preaching which Catholics think became embedded in the tradition of the Church, particularly in her liturgy.

2. Furthermore, unlike the development of geometry which is as far as we know is typically done by the unaided human intellect, the development of Catholic doctrine is claimed to be done by the human intellect guided by Holy Spirit.

3. Moreover, the Scriptures and the Tradition of the Church not only contain particular doctrinal axioms from which we can derive further propositions, but contain ways of reasoning or rules of inference that embody an understanding of how God deals with the world. Prominent among these is typology. In the New Testament and the Church's liturgy, we learn that God works through parallels. The people of Israel pass through the sea; Christians pass through baptism. Adam sins and from his sin comes death; Christ conquers sin and from his conquering sin comes life. The New Testament (Luke 24:27) says that all of the Old Testament scriptures tell us about Christ. Thus there may be substantive ways of reasoning embodied in Scripture, liturgy and theological practice, ways of reasoning that include typological reasoning. These ways of reasoning are, plainly, more than just formal rules of logic. They are based, rather, on an understanding of God as acting in certain ways (maybe with certain motives), as producing a certain kind of deeply interconnected history.

And new insights might well come from this. Christ corresponds in an important way to Adam; but Mary in the Church's understanding corresponds in an important way to Eve. Just as Eve was created without sin, so, too, Mary was created without original sin. Now it is true that prima facie one might have tried different typological correspondences--one might, for instance, make Mary's being conceived in sin be parallel-by-contrast to Eve's being sinless (as Christ's raising us is parallel-by-contrast to Adam's bringing death on us). Working out a deep understanding of the typology here, and connecting it with many other aspects of Christian doctrine, is going to be difficult. It may take centuries, thus, for the Church to settle on a particular understanding, e.g., to see that the parallel between the new creation in Christ and the old creation in Adam does in fact call not just for Christ the new Adam to be without original sin, but Mary the new Eve as well, but of course with her freedom from the weight of original sin flowing from Christ's redemption, just as our Church's freedom from the weight of original sin does.

Conclusion: It should be no surprise if from a very large body of axioms, which includes substantive rules of inference, one could derive many doctrines that one is individually surprised by.

Circles of justification

This is a fun little riddle, coming from a discussion with Dan Johnson. At t2 Mary believes q because she believes p. At t1 (t1<t2), she had come to believe p because she had believed q. No new evidence came in after t1 for p. Yet her beliefs that p and q are both justified and, indeed, knowledge. How could this be?

One solution: p and q are mathematical theorems. At t0 (t0<t1), Mary saw a proof of q. At t1, she saw that p easily follows from q. Between t1 and t2, Mary forgot all about q, the proof of q, and the fact that she derived p from q. She continued to know p, since we know mathematical theorems that we once had known the proofs of even if we do not remember these proofs. At t2, Mary realized that q easily follows from p, and came to believe q. Since she knew p, she now has knowledge of q.

Comments: This appears to involve a circularity in the order of justification, but only if we confuse the contents of beliefs with believings (or types of belief with belief tokens). Mary has three relevant, believings: (1) her believing between t0 until after t1 that q, (2) her believing starting at t1 that p, and (3) her new believing that q starting at t2. Here, (1) has independent justification; the justification of (2) depends on the justification of (1); the justification of (3) depends on the justification of (2). There is no real circularity.