I'm not sure what your tutor was getting at either. If your tutor meant that one can always enlarge a set of premises to make it an inconsistent set, that's obviously true: simply add the negation of one of the premises already in the set. If he meant that any axiom system with infinitely many premises (say, one that employs an axiom schema) is inconsistent, then there's no reason to believe that.

Thank you for your question. For a long while in the history of philosophy it was thought that what was conceivable was a good indication of what was possible. Descartes is a good example of this way of thinking, though he was careful to require that not any old conceiving of a thing showed it to be possible. Rather he required that the conceiving had to be "clear and distinct", meaning roughly that it had to pass the most stringent standards we can muster to make sure the conceiving is coherent (i.e., not subtly self-contradictory). In the middle of the 20th century this methodology began to break down. For instance, in the Sixties Hilary Putnam distinguished between concepts and properties, making clear that our concepts of things like gold may not reveal its true properties. Similarly, Kripke's notion a decade later of "natural kinds" made room for the possibility that what is "metaphysically possible" may not correspond to that is conceivable.

This issue is still a topic of intense debate. Some philosophers in the last decade or so have argued that conceivability considerations have *some* force in determining what is genuinely possible. For instance, see Frank Jackson's, From Metaphysics to Ethics: A Defense of Conceptual Analysis, Oxford University Press, 1997. I should mention that conceivability does seem an important tool for many fields, not just philosophy: for instance, physics uses thought experiments regularly. Those "experiments" are constrained by what we know of the laws of physics, but philosophers' thought experiments can take into account all we know also about the empirical world. Imagine, further, how hard it would be to reason in ethics or political philosophy without being about to construct thought experiments! The moral here is that conceivability considerations are not aimed at finding out about our minds, but rather are our attempts to use common sense, albeit fallibly, to find out about the world.

Mitch Green

I don't think that there is or could be a general principle that says that a paradox arising from idealizations will inevitably carry over (much less become worse) when the idealizations are relaxed. In some cases, the paradox will disappear when the idealizations are removed. In other cases, the paradox will persist (or become worse). There is no general rule here. It depends entirely on the details of the case.

For example, various paradoxes result in classical electromagnetic theory when pointlike charged particles are used. Point charges are a convenient idealization for many purposes, but the energy in such a field is undefined (the integral blows up). However, if we go to charge densities and extended charged bodies rather than point charges, these problems disappear. Likewise, in cosmology, Newtonian gravitational theory is afflicted with various paradoxes if we assume an infinite universe with a homogeneous, isotopic distribution of matter. Remove these idealizations and the problems go away.

These are intended merely as examples to show that sometimes paradoxes arise because of the idealization. In other cases, they arise despite the idealization. The matter can only be resolved in a case-by-case way, given the details of the particular paradox.

Well, it is certainly true that introducing unambiguous, very carefully defined, agreed terminology and having a perspicuous notation can make reasoning easier and make us less prone to common reasoning errors. To take the obvious example, mathematicians aren't just being awkward when they use a lot of symbolism and make very careful distinctions wrapped up into technical terms (and borrow from the languages of formal logic to make clear, for example, the 'scope' of their quantifiers). If proofs all had to be written out in unaugmented English, then we'd get lost following them, even in elementary high school algebra: and proof-discovery would be orders of difficulty harder.

I suppose we might say "mathematicians' English" -- meaning English augmented with their new definitions and notational devices -- is a new, better, language, more suited to (mathematical) reasoning than street English. But equally, we might say that it is just one part of a single inclusive language, modern English: it is just a part that is only learnt by those with certain specialist interests. But for present purposes I can't see that it really matters which of those descriptions you prefer. The key point remains that, yes, appropriate linguistic devices, e.g. sharply defined terms and a perspicuous symbolic notation, certainly can expedite reasoning and help us avoid error.

Why shouldn't two different chains of reasoning lead to one and the very same conclusion? Mathematicians often give different proofs of the same result. For example, Aigner and Ziegler's wonderful Proofs from the Book starts off with six proofs (chosen from many more) of the same proposition, i.e. Euclid's result that there is an infinite number of prime numbers. The very different routes to the same conclusion are illuminating, as they show up different connections between the fact that there is an infinite number of primes and other mathematical facts. But it is one and the same mathematical proposition that the different connecting proofs all home in on.

I've chosen a mathematical example first because of the question's emphasis on "logical reasoning". But the point generalizes to cover other sorts of grounds we might have for accepting a proposition. I take it that Jill is in the coffee bar, as she has just phoned me and told me she is waiting there for me right now. You take it that Jill is in the coffee bar as you've just glimpsed her through the window. Different routes to acquiring belief in one and the same proposition. Of course, Jill's phone call tells me other things too -- e.g. that she is cross to be kept waiting. And your glimpse tells you other things, e.g. that she is sitting in a window seat. I pick up one overall package of beliefs from my phone encounter: you pick up a different package from your glimpse. But that doesn't prevent one and the same proposition -- the proposition that she is in the coffee bar -- being in both packages. Why not?

I take it that by "bivalence", you mean the principle that every proposition is either true or false. And if we take that principle in unrestricted form---we really do mean every proposition---then, well, it's hard to see how it could fail to imply that the proposition expressed by "There will be a riot in London on 13 January 2076" is either true or false.

If you don't like that conclusion, then you have to abandon bivalence---or, perhaps, the claim that the sentence in question expresses a proposition, though that seems rather worse. But note that you do not have to abandon bivalence, so to speak, across the board. You might still think that every mathematical proposition is either true or false, or that every proposition about the past is either true or false, or.... Perhaps there is something special about the future here.

As you probably know, Michael Dummett argued that one way to understand debates over "realism" takes them to turn upon our attitude towards bivalence regarding propositions about the subject matter in question: So a view that gave up bivalence for statements about the future would be a form of "anti-realism" about the future.

Philosophy would be much easier if we could program such a machine -- and boring too. But it's not going to happen. For one thing, there's your interesting point that philosophical disputes can go very deep, so deep as to include disagreement about what the laws of logic, of correct inference, ought to be. Secondly, even for first-order classical logic, there simply is no computer that can decide whether any given inference is correct. (This is known as Church's Theorem and was proved by Alonzo Church in 1936.) Finally, there's the fact that evaluating the logical cogency of arguments is only a (small) part of the business of figuring out what to think about someone's argument in philosophy: at the very least, one must also understand and evaluate the assumptions to which the logical reasoning is applied.

See Question 442.

Philosophy departments like to tell themselves (and their funding bodies!) that the study of philosophy distinctively makes their students better all-round thinkers -- in the fashionable jargon, our courses deliver special "transferable skills".

Actually, that strikes me as really a rather unlikely claim (at least if it means any more than that our students grow up, get more mature, learn not to jump to conclusions, learn how to write well-presented coherently organized papers, etc., which happens with pretty much any serious academically rigorous degree course). Anyone who has sat through scores of departmental meetings, listening to various bunches of philosophers trying to muddle through organizing their affairs, often making a complete hash of it, knows perfectly well that -- outside their research work -- even the best philosophers are no better at thinking straight and keeping their eye on the ball than anyone else. (And after those departmental meetings, are the pub conversations about politics, say, noticably any sharper and more rational with philosophers than with other smart, well-informed, people? Certainly not.)

Doing a lot of philosophy sure improves your performance at philosophy, but otherwise doesn't seem to me to have especially good effects on "cognitive skills". But -- the question asks -- could it in fact have bad effects?

I remember the logician Geoffrey Hunter once telling me the following story. At the beginning of his elementary formal logic course, he'd give out a sheet of sample arguments, and ask students to tick off the ones they thought were good arguments. At the end of the course, he'd give out the same sheet. And average scores went down.

You can see why! Whereas at the beginning students had to think through the examples, doing the best they could, later they were tempted to be over-impressed with their shiny new half-understood formal tools, and apply them thoughtlessly, at least while in the logic classroom, forgetting (for example) all those warnings about translating "if" using the truth-functional horseshoe.

Does that mean introductory logic courses should come with a health warning? Well, no: for I bet the "damage" was temporary and superficial. Once outside the formal logic classroom, students quickly forget most of what we've said -- as again any philosophy teacher knows, the same old fallacies and confusions will still tend to turn up in their other work.

So I wouldn't worry too much. I'm fairly sure that philosophy courses have little distinctive impact for good or ill on general "cognitive skills" (which is fine by me, as the courses are, after all, supposed to be teaching philosophy). And I'm even more sure that such effects that they do have don't reach down to disturb the patterns of hard-wired quick-and-dirty cognitive processing which evolution has provided us with.

However, those are just my guesses. It would indeed be very interesting to have some careful, well-controlled, empirical research to appeal to, which tells us about the comparative impact that e.g. a couple of years as a philosophy major, has on general reasoning abilities. But I don't myself know of any work. Maybe some other panelist has some pointers?

You have certainly put your finger on a complex issue. One might say you've got a dragon by the tail.

First, I should call your attention to the fact that you've rendered his argument in two logically different ways. The first rendering is actually a valid form of deductive inference, not a fallacy. Philosophers, in their pretentious way, call it a modus tollens. The terms in which you've put it allow for this rendering:

1. If Not-P, then Not-Q.

2. Q.

3. Therefore, P.

And, by the way, that first rendering can also be restated in another valid form called a modus ponens:

1. If we have knowledge (Q), then we understand or acknowledge the law of cause and effect (P).

2. We have knowledge (Q).

3. Therefore, we understand or acknowledge the law of cause and effect (P).

There's a rather large issue lurking here, too, as to what "understanding" and "acknowledging" mean, how they're similar, how they're different. (See, for example, Stanley Cavell's, "Knowing and Acknowledging" in his book, Must We Mean What We Say? , Cambridge UP, 1976). But let's put that aside for the moment.

The second rendering you present (the way you put it in your very last sentence) is, take note, different from the first; and it is indeed fallacious. It's a fallacy known as affirming the consequent. That second rendering might be restated this way:

1. If we understand or acknowledge the law of cause and effect (P), then we have knowledge (Q).

2. We have knowledge (Q).

3. Therefore, we understand or acknowledge the law of cause and effect.

Okay, enough of the logic of your question. More importantly, I'd like to observe that while Kant's transcendental "deduction" of the idea that general causal judgments are true in a what philosophers call an "a priori" way might be called an argument, it's an argument of a rather unusual sort. I would caution you therefore about the risks of rendering Kant's argument in the simple deductive way you present it here. I think this sort of rendering masks the special way he thinks he is "proving" his case. Kant himself remarks that his proof is more akin to the judgment of a kind of legal tribunal than a proof of deductive reasoning. At the risk of butchering Kant too much, I'd suggest you consider Kant's "deduction" as a bit closer to this rendering:

1. A necessary condition for the possibility of knowledge of the natural world is the a priori truth of the claim that every event in the natural world has a natural cause. (In short, scientific knowledge is possible only if the law of cause and effect is a priori true.)

2. Let's proceed as if scientific knowledge is possible.

3. It's an a priori truth that every event in nature is caused by another event in nature (i.e. that the causal law is true).

Of course, that's a rough rendering, and I hope my colleagues will forgive me for its brutality; but note how my second premise is much more guarded than yours, and note how I emphasize that Kant is more interested in figuring out what the necessary conditions for the possibility of knowledge are than proving that we actually have knowledge. Anyway, I hope this helps you pursue a more advanced understanding of the issue.