Is it possible for a mathematical equation to both be fundamentally unsolvable and also have a correct answer?

I hope philosophers of math on the Panel will respond with more authority than I have. My understanding is that G ö del showed that arithmetic contains pairs of mutually contradictory statements neither one of which is provable within arithmetic. Assuming the standard logical law that exactly one of every pair of mutually contradictory statements is true, we get the result that some arithmetical truths are unprovable within arithmetic. I can't say whether those truths include statements to the effect that such-and-such is the solution to an equation, but if they do, and if their being unprovable within arithmetic makes the associated equations "fundamentally unsolvable," then the answer to your question is yes . Someone might reply that an unprovable arithmetical statement can't be true , but I think that would be to mistake truth for provability.

Would it be fair to say that philosophy is a manipulation of words, and that scientists deal with the relationship between language and extra-language observations? Thus "truth" would primarily be a language concept according to which consistency between words would exist. In the non-language (empirical) world truth would be infrequent because be empirical observations can rarely be one hundred percent verified.

To be candid, your question seems to embody some confusions. I'll try to address them in this reply. 1. I think it's fair to regard philosophy as the analysis (if you like, the logical manipulation) of concepts , although that view of philosophy is rejected by some philosophers. In any case, concepts can be expressed in any number of languages, so I wouldn't regard philosophy as the manipulation of words as such. 2. Scientists, as far as I can tell, don't in general examine the relationship between language and extralinguistic observations. Instead they try to explain or predict patterns of observations in as unified and elegant a way as they can manage. 3. I don't see how it follows ("Thus") from your first sentence that "truth [is] primarily a language concept according to which consistency between words would exist." First, what does "consistency between words" mean? Are "red" and "colorless" mutually inconsistent words because red and colorless are...

Would an omnipotent and omniscient being be bound by the laws of logic? If so, to what degree?

Yes. Completely. The tricky question is why . It's tempting to answer that necessarily everything is bound by the laws of logic because the alternative -- the claim that something isn't bound by the laws of logic -- is necessarily false. But, as I suggested in my reply to Question 4837 , no sense can be attached to the claim that something isn't bound by the laws of logic. So the claim can't be false , strictly speaking. Perhaps all we can assert is a wide-scope negation: it's not the case that something isn't bound by the laws of logic, just as it's not the case that @#$%^&*. Necessarily everything is bound by the laws of logic because the alternative is literally nonsense? I wish I had a better explanation!

In a classic episode of "Batman: the Animated Series" (called "Perchance to Dream"), Bruce Wayne discovers (spoiler alert) he is in a dream because he in unable to read a newspaper he picks up. At first there are some ordinary words in the headlines, but everything becomes a jumble of gibberish as he attempts to read more closely. He later explains his reasoning by claiming that reading is a function of the right side of the brain, while dreams come from the left. My first question is: is this just a clever plot device or does it hold any water neurologically? And second, if it were true, would it be an argument against I-could-be-dreaming-based skepticism? Finally, third, the dream Bruce is having is a pretty good one, involving lots of things he would like but can't have in the waking world. His murdered parents are alive again, he's going to marry a woman he loves, etc. Bruce says he can't accept it, however, because it "isn't real". If you grant that he could keep on living on the dream world,...

I'll try to answer the second of your three interesting questions. The proponent of the dream argument for skepticism (imagine rehearsing this argument to yourself) could say, "For all I know, this allegedly scientific claim about right and left hemispheres is merely more stuff from my dream; I can't tell that it's not. Even if it's a true claim, I can't know that it's true until I rule out the possibility that I'm merely dreaming it up." If so, then Bruce Wayne's reasoning wouldn't be an effective reply to the dream argument. This isn't to say that it's clear sailing for the dream argument. My view, for which I argue here , is that the dream argument is self-defeating unless it's no different from (and hence no improvement on) the evil demon argument.

Working off Kelsen, logic and rules of inference, as well as other rule based systems, are normative, "ought" based systems. If this is true, or even if it isn't, what reason do we have to take that logical rules are reasonable? In other words, why should one accept that rules of valid inference (of any system) as actually generating true responses from true premises?

To test a rule of inference, you can try to find counterexamples to it, cases in which the rule lets you derive a falsehood from true premises. Professor Vann McGee offered a well-known (and controversial) such attempt in this article . But there's no getting around rules of inference entirely. Even as you test one rule of inference you unavoidably rely on others. Because any attempt to answer the question "Why should we trust rules of inference at all?" will rely on reasoning, it will trust some rules of inference, whether or not those rules are made explicit in the reasoning. There's no way to get "outside" all rules of inference and see how they measure up against something more trustworthy than they are.

If Laws of logic are true or hold in all contexts, how can there be more one law? Do the two versions of De Mogan's laws differ? If so. how? Does the law of excluded middle differ from the law of non contradiction and from either version of De Morgans laws? Enoch

Notice that the same question arises in math, where the laws also hold no matter what. Arithmetic contains commutative laws of addition and of multiplication, associative laws of addition and multiplication, a distributive law of multiplication over addition, etc. Are those laws different? Their representations on the page certainly look different. I take it that you're asking, at bottom, how truths that hold in all possible worlds could count as distinct truths. The answer depends on how propositions are to be individuated , and here philosophers give various answers. On some theories, there's only one proposition that's true in all possible worlds, although there are indefinitely many sentences (some logical, some mathematical, some metaphysical) that express this single proposition. Other theories give a more fine-grained way of individuating propositions that allows for the existence of multiple propositions that are true in all possible worlds. You'll find more...

In predicate logic can we have valid arguments if we make an existential claim in our premises and not in the conclusion? In other words can we simply rename the existential quantifer to a "particular" quantifer or something of the sort? Does this particular quantifer always have to carry existential import?

If I understand your first question, the answer is no (unless the existential premise is superfluous). By an "existential claim," I take it you mean an existential generalization such as "There exists an x such that F x ," rather than a claim of the form "F a ," which implies an existential generalization. But you might wish to look into the rule of Existential Instantiation (or Existential Elimination in natural deduction systems); you'll find a brief summary of it here . I'm not sure I understand your second question. There are two ways of interpreting the universal and existential quantifiers: the objectual way and the substitutional way. I can't find a handy link to recommend, but if you search for discussions of those terms, you may find something relevant to your third question.

Me and my professor are disagreeing about the nature of logic. He claims that logic is prescribes norms for correct reasoning, and is thus normative. I claim that logic is governed by a few axioms (just like any in any other discipline, i.e. science) that one is asked to accept, and then follows deductively, free of any normative claims. My question is: which side is more sound? Thank you.

In this context, by "normative claims" I take it you mean claims that one ought to (or ought not to) do some particular thing. Can we get such claims out of principles of deductively valid inference? I think so. If you accept P, and you recognize that P implies Q, then there's a sense in which you ought to accept Q: you're logically and rationally committed to Q by propositions that you accept and recognize. If you accept Q, and you recognize that P implies Q, there's a sense in which you ought not to deduce P from those propositions alone: doing so would be fallacious. Now, you might say that the ought and ought not in those cases is only hypothetical: " If you want your deductive reasoning to be reliable, then you ought (or ought not)...." But I think the antecedent of that conditional (the "if" part) is easy to discharge. Plenty of people do want their deductive reasoning to be reliable, and so there's a sense in which such people really ought to use ...

Is it racist to use the word "niggardly," despite the word not being etymologically related to the notorious N-word?

It's not clear to me which of two questions you're asking: (a) Is it always racist to use the word "niggardly"? (b) Can it be racist to use the word "niggardly"? I'd answer "no" to (a). It's not racist, and it's accurate, to describe Ebenezer Scrooge (before his conversion) as a niggardly character. But suppose someone uses "niggardly," perhaps mistakenly thinking that it's related to the N-word, in order to express racial hatred. I think that counts as a racist use of "niggardly," so I'd answer "yes" to (b).