Argumentum ad Ignorantiam

Robert Larmer has an interesting little paper called Is there anything wrong with “God of the gaps” reasoning? on the proper characterization of the fallacy of argumentum ad ignorantiam, or appeal to ignorance. The "God of the gaps" charge is a very good place to examine that question, because it is probably the most widely known point in which it arises. It isn't the only such argument, though; for instance, many textbook characterizations of the fallacy would require us to put Dawkins's argument here in the same category. Much of Larmer's paper I find unconvincing, but he is quite right about how perversely uncritical some are when it comes to this sort of fallacy -- whose status of fallacy is not, in fact, even clear.

A common way of characterizing the fallacy is along these lines:

Statement p is unproved.
Therefore Not-p is true.

or

Statement not-p is unproved.
p is true.

But it's hard to see why this is fallacious; it is natural to read it as both defeasible and enthymematic. Walton, among others, has noted that as a matter of practical reasoning, this is often an excellent form of reasoning. (I have not determined that this gun is unloaded; it is reasonable to regard it as, for all practical purposes, a loaded gun unless it is shown to be otherwise.) But practical reasoning leaks into speculative inquiry; and as a purely speculative matter the above forms of reasoning can be quite reasonable. If I say, "There is no evidence that such-and-such drug has the effects attributed to it," I am not appealing to ignorance. I am appealing to the evidence of the inquiries themselves, the inquiries that have been made into the effects of the drug in question, and it stands or falls with the adequacy of these inquiries. Copi, in a typical hand-waving maneuver, tries to block this by saying that in such cases the argument is really based not on ignorance but on our knowledge that if the matter were to occur (e.g., if the drug were to have the effects attributed to it) it would be known. But this would save any supposed appeal to ignorance from being considered a fallacy. For surely they can all be interpreted as making the assumption that if p or not-p were true, this would be known or proven. Put them in the form of a modus tollens:

If p, it is proven.
It is unproven.
Therefore not-p.

If not-p, it is proven.
It is unproven.
Therefore p.

Thus the bar is so high that it seems absurd even to charge someone with the fallacy unless you have truly excellent reason to think they are basing their argument merely on a claim of general ignorance and nothing else, not even the assumption that we would discover it if it were true. But when are we ever going to encounter such an argument?

The Wikipedia article on the fallacy is about as close to perfectly incoherent as it could be. At one point it would seem to require us to say that every scientific conclusion commits the fallacy. For instance, it gives this as a nineteenth century instance:

The solar system must be younger than a million years because even if the sun was made of solid coal and oxygen it would have burned up within that time at the rate it generates heat.

Exactly where this example is from, I do not know. But on the suppositions presented in the article, this is an entirely reasonable argument that commits no fallacy; it provides a scientific reason for arriving at a conclusion, and one that was decent enough at the time. It errs in its assumptions about the nature of the sun, of course; but it doesn't appeal to ignorance in any way, shape, or form. Something doesn't count as an appeal to ignorance simply because there might be new evidence that could not have been pointed to as evidence at the time.

Walton points out, in an excellent discussion that the first classification of arguments that includes argumentum ad ignorantiam is given by Locke. It is one of "four sorts of arguments that men, in their reasonings with others, do ordinarily make use of to prevail on their assent; or at least so to awe them, as to silence their opposition." But Locke's argumentum ad ignorantiam is simply an argument by challenge: I argue that you should either accept my proof or provide a better one. As Locke notes, this argument does not advance us in knowledge, since knowledge can only come from "the nature of things themselves," not from my relative inability to avoid being outargued. But it does not follow from any of this that it is a fallacy, and Locke himself does not claim that it is. Locke's argumentum ad ignorantiam is inferior to the one form of argument that serves as a way to get knowledge, the argumentum ad iudicium, because it appeals to the interlocutor's ignorance (since you can't think of a better way than mine, shut up) rather than to his judgment. It shares this inferiority with the other two kinds of argument that don't tend toward knowledge, the argumentum ad verecundiam (which appeals to the interlocutor's modesty with regard to authority) and the argumentum ad hominem (which appeals to the interlocutor's own concessions). But it is not fallacious to appeal to experts (which is clearly a form of what Locke calls argumentum ad verecundiam), nor is it fallacious to show that if your opponent were consistent in his claims that he would agree with you (which is what Locke calls argumentum ad hominem); likewise, neither is it fallacious to appeal to the interlocutor's inability to have a better position than yours. It just doesn't give you "the nature of things themselves."

Walton always seems to assume that what Locke is describing is what later people would describe as the argumentum ad ignorantiam. It's not clear to me that this is so. Locke, I would suggest, is really talking about persuasion. His four sorts of argument are really four modes of rational debate, four ways in which you can persuade someone to assent to a position: by appealing to authorities and experts whose views they would or should respect (ad verecundiam), by pointing out their inability to argue for a better one (ad ignorantiam), by showing that what they have already conceded or would be willing to concede implies that position if they are to be consistent (ad hominem), and by giving them some direct evidence of the way things really are (ad iudicium). We all use these very often, and for good reason; they are essential parts of rational rhetoric, i.e., the rational means of persuading others. A longstanding attempt to salvage the notion of argumentum ad ignorantiam to see it as a fallacy of shifting the burden of proof. This clearly has the merits of both conveying something like what we usually mean by it and also conveying something like what Locke means by it. But it requires a very particular and very controvertible view of burden of proof even to hold that there are any fallacies related to burden of proof.

Leibniz had already pointed out in response to Locke's characterization of argumentum ad ignorantiam that there are cases where it is reasonable to accept a position until its contrary is proven. Whately, interestingly, claims in Elements of Logic that argumentum ad ignorantiam just means arguing with some kind of fallacy with an intent to deceive, and is therefore not a specific kind of fallacy at all. (Many informal fallacies are understood by Whately as only fallacious if there is an intent to deceive by substituting a not strictly relevant conclusion for the relevant one.) The same idea is found in Stock's Deductive Logic.

Which raises the interesting question of who started the use of phrase for arguments of the sort we usually mean by them. I find an obscure statement in McCosh's The Laws of Discursive Thought (1873) that the fallacy involves insisting that someone believe something "because he knows nothing to the contrary". He gives the right sort of examples, too: clairvoyance, animal magnetism, the cons of priests and pretenders, etc. But even earlier, William Dexter Wilson, in An Elementary Treatise of Logic (1856) says it "consists in proving that a given Proposition is true, because we know of no reason why it should not be true, or why the truth should be otherwise". His discussion has all the incoherence of modern discussions too: it's a fallacy, but not always, and the sort of reasoning is bad, except when it is not. (He does do better than Copi, however, in recognizing that cases of argumentum ad ignorantiam pretty standardly involve the assumption that if p were true we would or could know it.)

Of course, I think the 'God of the gaps' charge is often quite reasonable. I don't think it is so when it is understood as claiming that the argument commits argumentum ad ignorantiam. Rather, saying that an argument is a 'God of the gaps' argument involves saying that (1) there is a gap requiring some causal or explanatory factor; (2) God is posited as directly fulfilling the role of that explanatory factor; but (3) we have reason that we do not need, or in the end will not need, to posit God in this role at all. It's a claim that the argument is hasty, jumping to conclusions, by attributing something directly to divine intervention that we have reason to believe is directly attributable to natural causes, even if we are still in the process of determining exactly what they are. Understood this way, it has nothing whatsoever to do with the fallacy of appeal to ignorance, being a claim about the best way to go about filling the gap. Thus I do think Larmer is on the right track in suggesting that it is really about whether the conditions for a good search-based inquiry have been fulfilled.

L'Engle

Madeleine L'Engle died Thursday at the age of 88. A Wrinkle in Time and its sequels A Wind in the Door, A Swiftly Tilting Planet, and Many Waters (somehow I never managed to read the final sequel, An Acceptable Time, or the other books in the related O'Keefe series) were the inspiration of a great deal of thought about time, friendship, and the like in my childhood.

In this fateful hour,
I place all Heaven with its power,
And the sun with its brightness,
And the snow with its whiteness,
And the fire with all the strength it hath,
And the lightning with its rapid wrath,
And the winds with their swiftness along their path,
And the sea with its deepness,
And the rocks with their steepness,
And the earth with its starkness:
All these I place,
By God's almighty help and grace,
Between myself and the powers of darkness.

Omniscience and Necessity

There has been some lovely discussion over at Richard Brown's blog, "Philosophy Sucks!", devoted to the question of whether omniscience is compatible with free will. There is vigorous discussion in the comments. The posts so far are:

Plantinga on Free Will and Omniscience

(I Think) I Got It!

Third Time's the Charm

Re-Inventing the Wheel

Does God Know About Quantum Mechanics?

Much of the discussion has been less about God and more about to handle talk about possible worlds, but there are many other topics discussed in the posts and the comments on them.

Rationalists are Growing Rational

A Ballade of Suicide
G. K. Chesterton

The gallows in my garden, people say,
Is new and neat and adequately tall;
I tie the noose on in a knowing way
As one that knots his necktie for a ball;
But just as all the neighbours--on the wall--
Are drawing a long breath to shout "Hurray!"
The strangest whim has seized me. . . . After all
I think I will not hang myself to-day.

To-morrow is the time I get my pay--
My uncle's sword is hanging in the hall--
I see a little cloud all pink and grey--
Perhaps the rector's mother will not call--
I fancy that I heard from Mr. Gall
That mushrooms could be cooked another way--
I never read the works of Juvenal--
I think I will not hang myself to-day.

The world will have another washing-day;
The decadents decay; the pedants pall;
And H.G. Wells has found that children play,
And Bernard Shaw discovered that they squall,
Rationalists are growing rational--
And through thick woods one finds a stream astray
So secret that the very sky seems small--
I think I will not hang myself to-day.

ENVOI
Prince, I can hear the trumpet of Germinal,
The tumbrils toiling up the terrible way;
Even to-day your royal head may fall,
I think I will not hang myself to-day.

Notes and Links

* Susan Palwick has a lovely post on giving to the homeless.

* Terence Tao has a summary of the argument of the paper co-authored by Danica McKellar in mathematics. McKellar has been getting a considerable amount of publicity recently, since she's a former child actress who's published a paper in mathematics and, more recently, a book encouraging middle school girls to take an interest in mathematics.

* Apparently there are rumors that there will soon be a movie in the works based on Charles Williams's All Hallow's Eve. My favorite Williams novel is (far and away) The Place of the Lion, and AHE probably isn't even in my top five, but it's a decent enough story, and I can see why someone might think it would make an interesting movie.

* Pointing out mistakes that reporters make when talking about the Pope (or any major religious figure, for that matter) is usually like shooting fish in a barrel -- too easy to be anything more than tiresome. But occasionally there's a fun fish to shoot. Some Catholic blogs are noting a Reuters report on the Pope's September 2nd homily, which devoted some words to safeguarding creation. The report says:

Intentionally wearing green vestments, he spoke to a vast crowd of mostly young people sprawled over a massive hillside near the Adriatic city of Loreto on the day Italy's Catholic Church marks it annual Save Creation Day.

Strictly speaking this is certainly true; the Pope was intentionally wearing green vestments. But, as Amy Welborn points out:

Um, yeah. I’m hoping the Reuters reporter means “Intentionally wearing green vestments because that’s the liturgical color for Ordinary Time,” but I’m thinking, given the context into which the reference is woven…maybe not.

* An interesting post on the value-of-knowledge problem at Virtue Epistemology. I found it an interesting read even though I'm a skeptic about the value of knowledge: that is, while I have no doubt that there are particular cases in which knowledge is better than true belief, I don't think knowledge in general is better than true belief in general, nor that knowledge as such is better than true belief as such. (As I've noted before, there are many, many cases in which it is fairly obvious that knowledge is not more valuable than true belief, e.g., when the object of belief or knowledge is very trivial. Incidentally, it bugs me that this value problem, which I regard as relatively trivial, is usurping the name 'Meno Problem', which is already the name for a perfectly good and longstanding philosophical problem, namely, how we can come to know something that we don't know already and recognize that it was what we were trying to know.)

* Hurray! There's a website devoted to making the contribution of women to philosophy more widely known. It's only just getting up and running; almost none of my favorites are on the list: Lady Mary Shepherd, Catharine Trotter Cockburn, Mary Astell, Mary Wollstonecraft, Edith Stein, and more; I look forward to seeing what's put up about them. They do have Anne Conway up, and the patron saint of philosophers, Saint Catherine of Alexandria; as also the two Saint Macrinas, Julian of Norwich, and many more. There is a blog that posts notices of new biographies put up.

ADDED LATER:

* Rebecca has a quiz on the Trinity (answers here). I think #29 is a little ambiguous, but in all it's a great quiz.

* Stephen Matheson, a biologist at Calvin College, has two posts on common descent: here and here.

* There are several discussions going on about the nature of sexual identity, sparked by the Craig scandal; Jon Rowe has some comments, and Mark and Macht does, too, for instance (see the comments on these threads as well). A phrase that keeps coming up is 'sexual orientation'. I'm skeptical that there's any such thing. Obviously there are developed sexual tastes; likewise there are sexual sentiments, partly original and partly learned, that tend to arise for different people on different occasions; moreover, we have sexual self-images; and there's a habitual mix of these that constitute our standing (but not perfectly stable) sexual interests. But 'sexual orientation' is, first of all, a phrase people who classify themselves as heterosexual often use to put everyone else in a comfortably opposed category of sexuality, ignoring large overarching similarities; second of all, it is continually used to elide the difference among standing sexual interests, sexual self-image, sexual experimenting out of curiosity, occasional sexual dabblings due to other reasons, and the like. Both of these are symptoms that the phrase tends to have more use for rhetoric than for designating anything in particular in the real world. It also, it should be pointed out to many of those who use the phrase who would abhor the suggestion in any other context, tends to imply a sort of sexual teleology. It is in fact often explicitly used to do so, and those who use the term have a responsibility either to tell us what sexual teleology they think is involved, or justify their use of a teleological term for something non-teleological. 'Sexual orientation', in fact, is one of the many instances in thought about sexual life where we have a perverse tendency to use the obscure and perpetually shifting to explain the less obscure and more stable.

* A lecture on James Clerk Maxwell, mostly biographical, by Ian Hutchinson of MIT. (ht: Claw)

Notes Toward a Formal Typology of Argument V

In the first post I laid out in a rough way the notation for the typology.
In the second post I introduced the notion of attenuation and used it to establish hierarchies of arguments.
In the third post I introduced the notion of preclusion and used it to show how distinct hierarchies of arguments are interrelated.
In the fourth post I introduced the notion of candidacy in order to show one way the typology could be extended (and corrected an inaccuracy in earlier posts).

Here I'd just like to give an example. But first it may be useful to note a few points that help when applying the typology to real-world examples.

The first is that the only difference in bases that is important is a difference that affects the structure of the argument. Thus, one can have as one's bases a thousand separate pieces of evidence, and if they all together only form one reason for a conclusion, they still can be treated as a simple, one-base type of argument. It's only when one base is put forward as a reason for thinking that another base is a reason for drawing a conclusion that we get into multiple-base hierarchies.

The second is that in a one-base case, multiplying R's doesn't really affect anything. a : R_a(R_a(XT)T) is just a more verbose a : R_a(XT).

In Hume's Enquiry Concerning Human Understanding there is a famous passage in which Hume presents a counterexample to his own position, the missing shade of blue:

There is, however, one contradictory phenomenon, which may prove that it is not absolutely impossible for ideas to arise, independent of their correspondent impressions. I believe it will readily be allowed, that the several distinct ideas of colour, which enter by the eye, or those of sound, which are conveyed by the ear, are really different from each other; though, at the same time, resembling. Now if this be true of different colours, it must be no less so of the different shades of the same colour; and each shade produces a distinct idea, independent of the rest. For if this should be denied, it is possible, by the continual gradation of shades, to run a colour insensibly into what is most remote from it; and if you will not allow any of the means to be different, you cannot, without absurdity, deny the extremes to be the same. Suppose, therefore, a person to have enjoyed his sight for thirty years, and to have become perfectly acquainted with colours of all kinds except one particular shade of blue, for instance, which it never has been his fortune to meet with. Let all the different shades of that colour, except that single one, be placed before him, descending gradually from the deepest to the lightest; it is plain that he will perceive a blank, where that shade is wanting, and will be sensible that there is a greater distance in that place between the contiguous colour than in any other. Now I ask, whether it be possible for him, from his own imagination, to supply this deficiency, and raise up to himself the idea of that particular shade, though it had never been conveyed to him by his senses? I believe there are few but will be of opinion that he can: and this may serve as a proof that the simple ideas are not always, in every instance, derived from the correspondent impressions; though this instance is so singular, that it is scarcely worth our observing, and does not merit that for it alone we should alter our general maxim.

So let's take the background argument, which is a set of reasons for the claim that every simple idea has a corresponding impression whence it derives. This is an argument of type a : R_a(IT), where I is the claim about the relation between ideas and impressions. He then presents a contradictory phenomenon, the fact that we can conceive of the scenario of the missing shade of blue (mb), which "may prove that it is not absolutely impossible for ideas to arise, independent of their correspondent impressions," in other words, that it is possible for I to be false. There is some ambiguity here about whether the argument is supposed to be of type mb : R_mb(IMF) or the slightly weaker mb : R_mbM(IF). The phrase "not absolutely impossible" suggests to me that the intent is the weaker one; the idea appears to be that it is not in the strictest sense impossible for simple ideas to arise without simple impressions, not that it is possible given actual conditions. The type would then be mb : R_mbM(IF). But this is controversial; one might choose to emphasize the terms in which Hume ends his description of the scenario: "this may serve as a proof that the simple ideas are not always, in every instance, derived from the correspondent impression". This might well be taken to suggest that the argument is of type mb : R_mb(IMF). However, if we might also wish to emphasize the 'may' in "this may serve", in which case we have a different sort of argument altogether, namely, mb : MR_mb(IF). In this interpretation Hume is only admitting to the scenario's possibly being a reason for thinking that I is false. Thus we have three superficially similar but in fact rather different interpretations of the missing shade of blue:

mb : R_mbM(IF)
mb : R_mb(IMF)
mb : MR_mb(IF)

This sort of ambiguity is, in fact, very common in arguments made in modal terms that are not made very, very carefully. If the argument is of the first type, it is an argument that the missing shade of blue suggests that it is possible for I to be false (under some circumstances or other, which may or may not actually occur). If of the second, it says that I can be false under some conditions that might actually occur. And if of the third, it says that the missing shade of blue might be a reason for thinking that I is false. Which one we choose affects how we understand Hume's counter-reasoning, but it will be convenient to use the term B_mb to mean the exclusive disjunction: either the first, or the second, or the third.

After giving the scenario, and saying that it may be taken as proof that I is false, Hume then goes on to say the words that have so puzzled commentators, namely, "this instance is so singular, that it is scarcely worth our observing, and does not merit that for it alone we should alter our general maxim." This reasoning is, at least at first glance, of type s, mb : R_s(~B_mb(IF)T). That is, it's an argument that s is a reason for thinking the missing shade of blue is not a reason for thinking I false. Now, how does our understanding of B_mb affect our understanding of this line of reasoning?

If B_mb should be understood as mb : R_mbM(IF), then the idea would appear to be this: the missing shade of blue really is a legitimate reason for thinking that it is possible for I to be false. But the singularity of the case could mean, among other things, that it is unlikely, and perhaps not even possible, for I to be false under conditions that would actually obtain in the real world. Thus I can be taken to be true as a "general maxim" on the original argument Hume made.

If B_mb should be understood as mb : R_mb(IMF), then the point of mentioning the singularity of the case is not to say that it's so unusual that it might not ever occur, but that it's so unusual that even if it does occur it's not a good reason for taking I as Hume actually understands it to be false. For instance, the idea might be that Hume is only interested in what generally occurs "in the wild" (in a restricted application of the principle), not under highly artificial circumstances like that which the missing shade of blue would have to be.

If B_mb should be understood as mb : MR_mb(IF), then the singularity of the case would indicate that the scenario is not even a good candidate for an argument that I is false. This would seem to be contradicted by what Hume says immediately before this ("this may serve as a proof"), although, of course, that carries a typical Humean ambiguity: it may serve as a proof, but does Hume think it really does serve as one, or is he just observing that other reasonable persons might think it does? But it does seem less likely, so we can perhaps set aside the mb : MR_mb(IF) as an unlikely possibility.

Now, Hume is very often criticized for his response, but it is clear that many commentators take the argument to be of type s, mb : R_s(~R_mb(IMF)T) -- that is, they take it as a straightforward denial that the missing shade of blue is a counterexample to I, where it is taken as actually being able to occur. But it's important to note that we can take the conclusion of the argument to be not (IMF) but M(IF), in which case almost all the criticisms of Hume's treatment of the missing shade of blue miss the point, for they take it as saying that it can really occur, whereas Hume might not even be committing himself to it's being able to occur at all -- he might just be saying that there is some conceivable scenario (and for him conceivability is possibility) in which it might, without committing himself to the claim that this scenario is at all consistent with other truths about the world. That is, all he would be conceding is that I is not a necessary truth; and he is right that singularity can show this to be irrelevant to actually taking I to be true, i.e., can be an undercutting defeater for I.

This is all very crude, but I think you can get the idea. As I said in the first post, this whole thing is just an idea that came to me one Monday in late July and was worked out over a few days of sporadic thought. It's likely to need fine-tuning in a great many spots. Comments are very welcome.

Logical Epochs

I find it interesting that philosophers often have a tendency to speak of logic as going through two epochs: the traditional, or Aristotelian, in which syllogistic reigns, and the modern, governed by the predicate calculus. Of course, there are many more epochs than this, but it's easy for people to do it. Thus when we get into other epochs in logic, like the splendid era started by Boole and De Morgan, we find that there's often an attempt to assimilate it to one or the other. Englebretsen, working in a modern extension of the traditional term logic, regards Carroll as the last great traditional logician; it's not uncommon for people used to the predicate calculus to treat Boole et al. as a sort of prolegomena to Frege.

Of course, the great algebraic logicians -- Boole, De Morgan, Jevons, John Neville Keynes, Lewis Carroll, John Venn -- are not so easily shunted to one side or another. They certainly saw themselves as making a sharp break with traditional logic -- which they would have understood in terms of the old staple of Aldrich's Artis logicae compendium, published in 1691 but still the standard reference for logic up until Whately published his Elements of Logic in 1826 (and in many places in England for yet longer). (The reader of George Eliot might recall poor Maggie trying to read Tom's school edition of Aldrich, and not being able to divine how the work related to the living world.) No one can read Venn's Symbolic Logic without recognizing that this is in his mind a radically different approach than anything traditional logic has to offer. And there is something to it. He regards the traditional emphasis on particular propositions (I and O) to be largely misguided -- one can put the traditional syllogistic, with I and O so prominent, in Boolean terms, but that is merely one permutation, and not an essential feature at all; logicians only work with universal propositions, and only deign to bother with particular propositions insofar as these can be considered incomplete universal propositions. One also finds that he reviewed Frege's work; it was a scathingly contemptuous review. He regarded Frege's notation as clumsy and generally not well-suited to application (which everyone agrees with) and his logical work to be far inferior to the algebraic logic being done in the wake of Boole (which is rather more controversial).

It's also true, I think, that there tends to be a complete difference in logical approach. It's dangerous to lump all the pre-Boolean work into the category 'traditional logic'; that ignores the fact that, for instance, the Ramists or the Cartesians tried to incite revolutions against what they perceived as the Aristotelian oppressor. And the sort of traditional logic the algebraic logicians would have known would have been a mix of simplified school manuals and Whately's adulterated revival version. But we can perhaps say to some extent that whereas logicians in the Aristotelian mode are intently focused on demonstration, the Booleans were not. They are very helpful in telling us what they are obsessed with, and it is not demonstration at all. Jevons puts it succinctly in his Philosophical Transactions: "Boole first put forth the problem of Logical Science in its complete generality: Given certain logical premisses or conditions, to determine the description of any class of objects under those conditions." As Keynes puts it in a passage in Studies and Exercises in Logic, where he quotes Jevons on this point:

Given any number of universal propositions involving any number of terms, to determine what is all the information they jointly afford with regard to any given term or combination of terms.

This is closely related to another obsession of algebraic logicians, the finding and eliminating of superfluous premises. It is not something we generally do much any more, at least as anything more than an occasional technical exercise; but for them the whole point of logical analysis is to determine precisely the most the premises give you and to determine precisely how you can get your conclusion without going over the same ground twice. But the predicate calculus is not used this way; I imagine that if there's an obsession it has brought about, it is identified by Quine in that passage in which he says that the whole point of logical grammar is to facilitate the tracing of truth conditions. One sees this difference in a number of ways they approach things. The algebraic logicians are generally dismissive of any attempt to reason about objects without specifying a domain of discourse; any symbolic system, for instance, that does not clearly specify a universe of discourse is regarded, by that very fact, as deficient. This does, in fact, tie in with their interest in the information afforded by terms in universal propositions. It is often difficult to get people trained in predicate calculus even to take universes of discourse seriously.

Logic, then, at least to a limited extent is not merely a set of formal considerations; it involves an approach, and what you are trying to do plays a key role in how the whole operation works. What counts as success, what counts as failure, what counts as even important, shifts depending on your view of the ends of logical thinking.