You asked, "Is there at least a generally accepted solution to this paradox?" Not by a long shot! The paradox of the heap (and its cousins that use other vague concepts) is in my opinion one of the greatest unsolved intellectual problems. It has generated a huge philosophical literature, and it's very much a topic of current philosophical debate, but I have yet to see a proposed solution that even comes close to being satisfactory. For starters, you might take a look at these entries from the Stanford Encyclopedia of Philosophy:

SEP, "Sorites Paradox"
SEP, "Vagueness"

Best wishes as you work your way through this daunting -- but inescapable -- problem! I think you'll find it repays your careful thought even if you don't end up much closer to a satisfying solution.

Great question that gets to the heart of a current debate! If you have a very narrow concept of logic (in which logic only refers to the laws of identity, non-contradiction, and the law of excluded middle) and if your notion of observation is again narrow perhaps only allowing in empirical data then perhaps it is neither logical nor illogical to mourn the death of someone. BUT, you may have a broader concept of observation. For example, in your question you refer to "a loved one." Can one observe the fact that a person is worthy of love or should be loved? I personally think one can. In that case, it would be quite logical (you would be acting with consistency) for you to act in a way that is appropriate when one's beloved one dies. On this expanded front, imagine you truly love Skippy and desire her or his happiness; that is, you believe it would be good for Skippy to be happy and bad if Skippy were to die before fulfilling the desires of his or her heart. Then, surely, it appears you should mourn Skippy's death. Matters may turn out otherwise, however, if you deeply restrict concepts like love, logic, and observation. I suggest the more open approach is the better one in that it captures more fully the way in which our experience is saturated with values that call for our response. You might check out Parfit's extraordinary two volume work On What Matters for a look at the issues and why there is some dispute today among philosophers on the fact/value distinction.

Rather than offer a response to this excellent question, let me just refer you to a paper whcih essentially raises and discusses the very same problem: Susan Haack's "A Justification of Deduction," from the journal Mind in 1976 (try vol 85, n. 337 I believe). Also, Lewis Carroll (as in "Alice in Wonderland" has a similar, more fun version of it -- "What the Tortoise said to Achilles" -- also in Mind, in 1895 or so ... Check them out!
ap

It goes by another name, sometimes "argument by cases" or "argument by dilemma" or "the disjunctive syllogism". The basic rule is:

A ∨ B
~A
∴ B

Obviously, this can be extended to any number of disjuncts, e,g.:

A ∨ B ∨ C ∨ D
~A ∧ ~B ∧ ~C
∴ D

So the disjuncts represent the possibilities you have before you, and the negations represent your ruling out all but one of them.

There are quantificational versions as well, e.g.:

∀x(Fx → x = a ∨ x = b ∨ x = c)
~Fa ∧ ~Fb
∴ Fc

This would normally be derived from one of the propositional versions.

Great set of thoughts, here. But maybe one quick mode of response is to remark that much depends on just what you take the word "cause" to mean. You could take it to mean something like this: "x causes y" = "y if and only if x", as you've suggested. Then, granting that both cases above are cases fulfilling the "if and only if", sure, giving birth would count as a cause of the later death. But now two things. (1) Why should "cause" mean precisely that? Wouldn't it be enough if the x reliably yielded the y, even if things other than x could yield the y too? (i.e. couldn't you drop the 'only if' part, so 'x causes y' would mean 'if x, then, y', even if it might also be true (say) that 'if z, then y'?) Going this route would preserve your intuition that both cases above are cases of causation, but focus on whether your particular definition is the best one. (2) Perhaps more importantly, though, one might examine the 'pragmatics' of causation -- how people actually use the word, different from how very precise philosophers or scientists might define it. So, for example, we often restrict the word 'cause' not just to every factor which may be necessary or sufficient or both for an effect, but to the most salient factors, the most explanatorily relevant ones, the most proximate ones, etc. So, you strike a match and it lights; strictly speaking many things are at least necessary for that (the presence of oxygen, the existence of the match, the laws of physics, etc.) but we often say 'the striking caused the lighting', even though all those other factors were necessary. Indeed the striking was neither necessary nor sufficient for the lighting -- the match could have been lit other ways, and striking on its own (without oxygen etc) wouldn't light. So our ordinary use of 'cause' is far looser than some technical philosophical definition. So the question for you is: how, ultimately, are you going to use the word 'cause', and are you justified in choosing that use in light of competing uses?

hope that's useful ...

ap

Let's assume it's true that "If P, then Q". The conditional claim that you imagine being inferred from this has the structure "If not-P, then not-Q". [Not quite: I don't think the negation of "we lower standards" is "we raise standards". One way in which we might fail to lower standards is to keep them fixed.] This is indeed an incorrect inference. The first conditional claims that P is a sufficient condition for Q. While the second claims that P is a necessary condition for Q. And the latter claim simply doesn't follow from the former. For instance, it's true that if Rex is a dog, then Rex is a mammal. (Being a dog is a sufficient condition for being a mammal.) But this does not imply the false claim that if Rex is not a dog, then Rex is not a mammal. (Being a dog is not a necessary condition for being a mammal.) This fallacy is sometimes called The Fallacy of Denying the Antecedent. ("P" is called the antecedent of the first conditional claim above.)

I think your argument is logically valid--that is, IF the premises were true, then the conclusion would be true. And I don't think it commits any formal or informal fallacies (except perhaps equivocation in the sense I'll explain shortly).

The problem is that it is unsound, because it has at least one false premise; hence the conclusion is not "made true" by the premises. Premise 3 is false. The government does not tell us not to kill no matter what. As you point out, it tells us not to break specific laws against specific types of killing. Typically, citizens are not breaking the law (and are morally justified) in killing in self-defense or to protect others from an immediate and deadly threat. And (legal) killing in war and use of the death penalty (where it is legal) are also not forms of killing the government tells us not to commit.

Now, we may have reasons to think that some or even all killing in war is morally problematic and even more reasons to think the death penalty is morally wrong. And we have greatly narrowed the scope of such legalized killings over time (in the U.S. and even more so, in other industrialized nations, most of which, for instance, have made the death penalty illegal). And we may believe that it is hypocritical to say some killing is OK but not others (though almost no one, perhaps Jesus excluded, suggests that you cannot kill, if necessary, in self-defense). But I don't think that the government is "saying one thing but doing another" in these cases, because the government, just like most of us, does not treat all killings as the same thing (hence the equivocation in the use of "killing").

I don't know that I can settle anything. The dispute you are having is one philosophers today have generally. Some people think logic is normative, in that it prescribes rules concerning how one should think, or reason; other people think logic is purely descriptive, and that it simply tells us something about the notion of implication or validity.

One reason people often given against the normative interpretation is that the norms logic provides just seem like bad ones. For example, it was once argued that, since logic tells us that A and ~A imply anything you like, then logic would be telling us that, if you reach a contradiction, you should infer that the moon is made of cheese; but, of course, what you should actually do is figure out what went wrong and give up one of the contradictory beliefs. The obvious reply, though, is that this is too simple a conception of what the norms logic prescribes are. It assumes, in particular, that if A implies B, then it is a norm that, if one thinks A, one should infer B. But maybe the norm is that, if one thinks A, then one ought either to infer B or to give up A. Logic won't tell you which one to do, but it demands you do one or the other.

A deeper concern is that reasoning itself might be more involved with probability than logic allows. In that case, the norms of reasoning would presumably come from probability theory, and classical two valued logic would have the wrong subject matter. (Let me insert a plug here for my colleague David Christensen's book Putting Logic in Its Place and suggest you read the entry on Bayesian epistemology at the Stanford Encyclopedia.) On the other hand, however, there are known ways of essentially deriving probability theories from logical theories, so two-valued logic would represent a sort of idealization to the case of absolute certainty of the norms that govern reasoning, which actually proceeds, most always, under uncertainty.

One sure way to prove invalidity is to describe a possible case where the premises of an argument are true and the conclusion false. To make things a bit more plausible, let's change the example slightly. The following is valid:

"q, not-p implies not-q, therefore p"

I pick this example because this argument (closely related to modus ponens) is one that people have a little more trouble seeing, or so my experience teaching logic suggests. So there could be and likely are cases where a person judges that q, and judges that not-p implies not-q, but has trouble with the logical leap and therefore fails to judge that p. That's a counterexample to the argument you're interested in. We have someone who judges the premises of a valid argument to be true but doesn't judge the conclusion to be true.

This isn't surprising. To judge something is (putting it a bit crudely) to be in a certain state of mind toward it. Being in the "judges that" state of mind toward the premises of an argument doesn't guarantee that someone will be in the same state of mind toward the conclusion, even if the conclusion happens to follow. Consider some argument with 10 premises P(1), P(2)...P(10) and a conclusion X. And suppose that X really does follow from P(1)... P(10). Your suggestion seems to tell us that anyone who judges the premises to be true will actually judge the conclusion to be true, even though, we'll suppose, the reasoning required to get from premises to conclusion is subtle and complex.

If you think about it, it's not at all unusual for people to miss seeing what their beliefs imply. Math is a particularly obvious case; getting someone to accept the principles of Euclidean geometry doesn't guarantee that they'll simply judge all the theorems to be true. But math is by no means the only case.

Another way to to put it: the problematic argument you judge to be valid needs another premise. A general version of the premise would be something like this:

Whenever an argument is valid and I judge that its premises are true, I always judge that its conclusion is true as well.

That's false for any human "I."

If by 'logical' you mean 'a decent argument can be constructed of this form' then i would say the answer is yes -- but if you mean 'an absolutely convincing argument ...' then, well, you don't find too many of those anywhere in philosophy -- my favorite version of the kind of argument that Allen Stairs mentions is some version of the fine-tuning argument -- which observes how perfectly fine-tuned features of the universe seem to be, such that they could easily have been otherwise, and yet had they been otherwise then human life (conscious, rational, moral life) would not have been possible -- and goes from there to argue that it is reasonable to think this didn't occur by chance -- a good source on this topic would be any of Paul Davies' recent books ...

best, ap