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Why is it important to study logic in philosophy? One answer might be that logic teaches you correct reasoning, but that is not something that is unique to philosophy, as it's important in other fields as well (e.g. history, economics, physics, etc.), and those usually do not include any explicit study of logic.

In my experience, philosophy courses take the explicit, self-conscious formulation and evaluation of arguments (i.e., reasoning) more seriously than any other courses of study, with the possible exception of those math courses that emphasize proofs. Moreover, the breadth and depth of philosophical problems exceed those encountered in math. Therein lie the advantages of philosophy courses as compared to, say, math or economics courses. If you pursue philosophy, I think you'll discover that the standards of argumentative rigor expected in philosophy courses surpass -- sometimes by far -- the standards of rigor expected in any courses outside of math, and again they're applied to a much more varied, and often deeper, set of questions.

Hey Philosopher folk: Do you know of any viable or at least well-examined arguments ever proposed that conclude that one murder (or some equivalent malfeasance) is no better nor worse than 8 million murders? Or generally, that multiple instances of a wrongdoing have no greater or lesser value of any kind, apart from numerical? If not, could anyone conceive of a possible argument for this? Please note, I am not a serial killer or mass murderer, this question just arose in a debate about an unrelated topic.

Well, I'm glad to hear you are not a murderer. If you were, I would argue that it is worse to kill greater numbers of people like this:
1. If act or outcome A is morally wrong, then A x n (n number of As) is more morally wrong than A. [stronger version might say A x n is n times morally worse.]
2. Murder is morally wrong.
3. So, n murders are morally worse than one murder. [Or any greater number of murders is worse than just one, perhaps n times worse.]

Like most good arguments, this one just puts things in a good form for us to be able to consider the premises. It sounds as if you (like me) accept premise 2. So, what justifies premise 1? The easiest way to justify it is if one is a utilitarian (or other consequentialist) who measures wrongness in terms of bad consequences or outcomes. So, if one murder causes X amount of bad consequences (e.g., suffering, loss of potential flourishing for victim, etc.), then n murders would cause (roughly) nX bad consequences. And it would be morally worse [n times worse] for that reason.

But on just about any moral theory, more wrong acts or outcomes is worse than fewer. If a Kantian says one murder is wrong because it violates the categorical imperative (it cannot be universalized, and it treats the victim as a means not an end), then she will also be able to say that Hitler's (helping to cause) 8 million murders is much worse, because he has done something wrong many more times.

Maybe the basic support for premise 1 is a powerful moral intuition that seems hard to deny.

If there is a category "Empty Set" it has to have the property "emptiness". It must have this property that separates it from every other set. Thus it is not propertyless - contradiction?

I don't see a contradiction here any more than I did back at Question 26649, which is nearly identical. Yes, the empty set has the property of being empty and is the only set having that property. But the emptiness of the empty set doesn't imply that the empty set has no properties. On the contrary, it has the property of being empty, being a set, being an abstract object, being distinct from Mars, being referred to in this answer, etc. Why would anyone think that the empty set must lack all properties?

Doesn't trying to demonstrate how we know anything beg the question?

It needn't. Like Descartes, you might try to demonstrate a priori that you possess perceptual (i.e., external-world) knowledge. Your demonstration needn't presume perceptual knowledge in the course of demonstrating that you possess perceptual knowledge. Therefore, your demonstration needn't beg the question of whether you have perceptual knowledge in the first place. Most philosophers, I think, regard all such demonstrations (including Descartes's) as failures, but I don't see any reason to think that all such demonstrations must fail because they beg the question.

Consider a more interesting case. Suppose I analyze knowledge as true belief produced by a reliable mechanism, i.e., a mechanism that yields far more true than false beliefs in the conditions in which it's typically used. A skeptic then challenges me to show that some perceptual belief I regard as knowledge, such as my belief that I have hands, was in fact produced by a reliable mechanism. In response, I offer empirical evidence in favor of my belief: others verify that I have hands; I clap my hands; I shake the skeptic's hand; I cite other outputs of my perceptual belief-forming mechanism, etc. The skeptic then protests that I'm begging the question because my method of verification simply assumes that I have perceptual knowledge: I simply assume that the evidence I offer was reliably obtained.

I think the skeptic's protest is unfair. It's unfair to ask me to show that my perceptual belief-forming mechanism is reliable while disallowing me the very means I would need to show it, namely, data that I obtain by perception. Alternatively, the skeptic could try to show that my belief-forming mechanisms are not reliable, but skeptics seldom if ever try to show that. Or the skeptic could reject my analysis of the concept of knowledge, but then the skeptic would have to offer grounds for rejecting it, which skeptics seldom if ever do.

Consider the mathematical number Pi. It is a number that extends numerically into infinity, it has no end and has no repeating pattern to its digits. Currently we have computers that can calculate Pi out to many thousands of digits but at a certain point we reach a limit. Beyond that limit those numbers are unknown and essentially do not exist until they are observed. With that in mind, my question is this, if we could create a more powerful computer that could continue to calculate Pi beyond the current limit, and we started at exactly the same time to compute Pi out beyond the current limit on two identical computers, would we observe the computers generating the same numbers in sequence. If this is the case would that not infer that reality is deterministic in that unobserved and unknown numbers only become “real” upon being observed and that if identical numbers are generated those numbers have been, somehow, predetermined. Alternatively, if our reality was non-deterministic would that not mean that the two computers would generate potentially different numbers at each iteration as it moved forward into unobserved infinity inferring that unobserved reality is not set and therefore we live in a reality defined by free will?

You're no doubt right that any computers we happen to have available will only compute π to a finite number of digits, though as far as I know, there's nothing to stop a properly-designed computer from keeping up the calculation indefinitely (or until it wears out.) But you add this:

"Beyond that limit those numbers are unknown and essentially do not exist until they are observed"

Why is that? Let's suppose, for argument's sake, that we'll never build a computer that gets past the quadrillionth entry in the list of digits in π. Why would than mean that there's no fact of the matter about what the quadrillion-and-first digit is? What does a computer's having calculated it or (at least as puzzling) somebody having actually seen the answer have anything to do with whether there's a fact of the matter?

To be a bit more concrete: the quadrillion-and-first digit in the decimal expansion of π is either 7 or it isn't. If it's 7, it's 7 whether anyone ever verifies that or not. If it's not 7, then it's something else whether or not anyone every figures out what. It would take a lot of arguing to give a reason why we should think otherwise.

This is related to another of your questions. Yes: if two properly-programmed, properly-functioning computers kept spitting out the digits in π, they would print out the same digit when they got to the quadrillion-and-first entry. π is the ratio of the radius of a circle to its circumference. This number is the same for all circles not as a matter of incidental empirical fact but because of what it is to be a circle. That ratio is a specific number, and if two computers disagree about one of the digits, at least one computer is wrong.

But this doesn't help us with the question of determinism. Let's suppose that the world is indeterministic. (For all any of us knows, it is.) This doesn't mean that every physical process is also indeterministic. It just means (roughly) that from the laws and the total state of the world at one time, the total state of the world at other times doesn't follow. But the word "total" matters here. Even if quantum processes are indeterministic at some level, the output of a computer running certain sorts of programs isn't. If a computer fits the requirements for being a Turing machine, its output is deterministic. Of course, any physical machine can break down, and it could be that what makes some particular machine crap out has a chance element. But that doesn't mean that all particular physical processes are indeterministic and in any case, doesn't make mathematics mushy.

Finally, on free will. The connection between free will and physical determinism is actually not as simple as it seems. If you're interested in thinking about that, there is a recent book by Jenann Ismael called How Physics Makes Us Free. It's accessible, engaging and rigorously argued. I recommend it highly.

Hi...I'd like to begin reading Hume. Should I begin with the Treatise or the Enquiry?

Well, there are two Enquiries, corresponding to the first and the third books of the Treatise. And I'm sure everyone will have her own strategy for reading Hume. My own opinion is that you can't really appreciate the Enquiries until you see how much is behind them; they're too smooth and polished. So I would recommend starting with the Treatise, but not reading it straight through from beginning to end, and not getting too bogged down in the minutiae. Very roughly, I would recommend reading Book 1 of the Treatise relatively quickly to get an overview of the argument, without attempting to be too precise about it. Then I would skip to Book 3 and do the same, though this one is a bit harder to grasp without attending to the details. It is fashionable these days to claim that the long-neglected Book 2 is just as important etc. as Books 1 and 3, but as a way in to Hume I think you'll find Books 1 and/or 3 more accessible. Also, depending on which you are more interested in (Book 1 if you're more into metaphysics and epistemology, Book 3 if you're more into social and political philosophy), there are the corresponding Enquiries to give you a nice overview from a somewhat different viewpoint, just as the Prolegomena gives you a nice overview of Kant's 1st Critique from a somewhat unexpected point of view.

Also there is some secondary literature that can really help to inform your reading. Among the things I have found helpful are Mossner's old biography, which still has merits even though James Harris's recent intellectual biography has in some respects superseded it (and is also very much worth reading). And depending on whether you're more into the epistemological and metaphilosophical aspects or the social and political aspects, you should look at things like Graciela de Pierris on the former and Russell Hardin on the latter. Also, if you're really focused and committed, you might consider looking at Duncan Forbes on Hume's philosophical politics (a challenging book even then). And there are plenty of new things that consider particular aspects. Look in the journals for new articles about Hume, for instance, and just start reading whatever sounds interesting to you from the title.

Only after that should you then go back and start reading the Treatise from the beginning, line by line, carefully. Again, depending on your interests, focus on Book 1 or Book 3 (or, by then, if you insist, on Book 2). Best of luck, it's worth sticking with, as you'll soon figure out once you get over the initial hurdles!

Are there any true contradictions?

None that I can think of, including none of the candidates that I've seen offered by "dialetheists" (i.e., philosophers who say that some contradictions are true). If you have any promising candidates, please let us know!

If there is a category "Empty Set" it has to have the property "nothingness". Thus it is not propertyless - contradiction?

As far as I can see, the definitive property of the empty set is not nothingness but instead emptiness: It's the one and only set having (containing, possessing) no members at all. The empty set can be empty, in that sense, without itself being nothing. So I see no threat of contradiction here. Indeed, the empty set can belong to a non-empty set, such as the set { { } } , which couldn't happen if the empty set were nothing.

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