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...

Since I answered the original question, I will try also to answer this one. We need to reconsider the phrase "social construction and nothing more", or at least to what you take to be the implications of such a description, that somehow what is socially constructed isn't real. One would need a lot of argument to establish that conclusion. Prima facie, socially constituted facts are no less real than biological or anatomical facts; they are just different. Consider, e.g., facts about political and legal authority. Surely these are socially constituted, but I would not suggest you tell a military tribunal that you can't be guilty of disobeying an order from a superior because social facts are unreal. That should answer question (1), I hope. Similarly, socially constituted facts matter to people every bit as much (and in some cases more) than biological facts. As I pointed out in response to the previous question, the mere fact that gender is a social (not merely anatomical) matter does not imply...

Logicians have long distinguished between "indicative" and "subjunctive" conditionals. The terminology reflects a difference, in English, in the grammatical "mood" of the antecedent and consequent. So we have: If Kennedy was not assassinated, he is living is Columbia. If Kennedy were not assassinated, he would be living in Columbia. The view to which you refer, that a conditional with a false antecedent is always true, has certainly been held, but only about indicative conditionals. So (1), on this view, is true if Kennedy was, as we all suppose, assassinated. But it is an entirely different claim that (2) is true simply because Kennedy was assassinated, and I know of no logician who has ever held that view. This is largely because some subjunctive conditionals, such as (2), are precisely intended to report on what would have happened had things been other than we know (or at least presume) they are. Since, as you say, it would be pointless to utter such conditionals, which are known as...

"...[A]ll sciences are products of our minds. They are our constructions, as are most of the physical objects in our immediate worlds." That is no doubt true, but it misses a crucial point: Scientific theories are of course human creations. But what those theories are about are (generally) not human creations. People do not make quarks, atoms, or molecules, fields, stars or galaxies, bacteria, birds, or insects, etc, etc, etc. Nor is it up to us whether the theories we invent are true. And even when physicists discuss colliding billiard balls, the fact that the balls were made by us is neither here nor there. They are external objects, and it is not up to us how they will behave. Much the same is true of mathematical objects. I see no reason whatsoever to believe that numbers are a human creation, any more than tectonic plates are. And it is just a confusion to think that a circle is an idea. Surely whatever ideas I have are in my mind. Is mathematics supposed to be about the contents of my...

While I agree with Oliver's judgements, I also think many people are genuinely puzzled about how someone could feel as if they had the wrong sort of body, so let me try to say a few things that might help to clarify it. I should say, however, that I am no expert on these issues, and so I'm sure to get some of the established terminology wrong. Let's start with a different sort of example. I am of Irish descent. But I do not think of myself as Irish-American. From my point of view, the fact that many of my ancestors lived in Ireland is just that: a fact about my family's past. For other people, however, being of Irish descent is very important to their sense of who they are. They value Irish traditions and customs, participate in Irish celebrations, and so forth. It is, as we say, part of their identity to be Irish-American. And so, if someone tells a rude joke about "the Irish", I wouldn't really take it personally; I'd just dismiss that person as a jerk. Someone who identifies as Irish-American would,...

It's true that multiplication can be defined in terms of addition, but the crucial question is: What logical resources are needed for that definition? The usual definition would be in terms of repeated addition, which means that the definition is (primitive) recursive. And now the point is that the theory we're discussing, which is sometimes known as "Pressburger arithmetic", doesn't have the resources needed for that sort of definition. So, in fact, this theory does not provide everything that's needed for defining multiplication, only some of it. Gödel does use primitive recursion to define various functions in the course of his proof. Indeed, as the proof is often presented, the central lemma is that every primitive recursive function is "representable" in PA (or whatever theory we're discussing). The details are not essential here, except that the construction crucially depends upon the presence of both addition and multiplication. (Yes: If you drop addition, then the first...

So far as I know, there is no one who holds quite this view. The reason is very simple. We say that an inference is logically valid just in case, whenever the premises are true, so must the conclusion be true. So if one starts with some true premises and "logically derives" some conclusion, then the conclusion has to be true, simply by definition. This assumes, of course, that our logical derivation is correct: that we haven't made mistakes, that the inferences on which we're depending really are logically valid, and so forth. Now, all of that said, philosophers can and do disagree about what inferences really are logically valid. So you might have thought that the inference from "If I go to the movie, then, if I go to the movie, then I'll have a good time" to "If I go to the movie, then I'll have a good time" is logically valid. But there are philosophers who deny that it is. In this particular case, they think, there are additional hidden premises that can be used to make it valid. But all by...

One way to answer this question, I think, would be to consider the history of science. Ptolemy, for example, believed that the earth was at the center of the universe, and that the sun and other planets revolved around it in roughly circular orbits, except for "eccentricities" accounting for which was much of what astronomers did in those days. Copernicus corrected part of that, holding instead that the sun was at the center of the universe and that the Earth and other planets revolved around it, with only the Moon (now not considered a planet) revolving around the Earth. But Copernicus too thought that the orbits of the planets were roughly circular, except for eccentricities By the time of Descartes, it was realized that the sun is not at the center of the universe, but it one star among many, though Descartes did think the sun was at the center of (what we would now call) the solar system. Kepler would later replace the view that the orbits of the planets are circular with the much more nearly...

As Nicholas said in response to the other question, there are questions to be asked about what is appropriate and inappropriate behavior in the workplace. And, while there are companies that prohibit co-workers from dating, most do not, which is simply to say that sexual harassment policies are not in general intended to prohibit all sexual interaction between co-workers, but only such interaction as, first, is unwelcome or unwanted and, second, constitutes a form of harassment. Even unwelcome sexual attention, by itself, does not constitute harassment, according to the definition promulgated by the Equal Employment Opportunity Commission, but only if: submission to such conduct is made either explicitly or implicitly a term or condition of an individual's employment, submission to or rejection of such conduct by an individual is used as the basis for employment decisions affecting such individuals, or such conduct has the purpose or effect of unreasonably interfering with an...

I don't know anything about Hegel, but I have several friends who reject the Law of Non-contradiction, and they're all perfectly respectable analytic philosophers, with lots of friends who are also analytic philosophers. So I doubt that the claim that Hegel rejects the Law of Non-contradiction, in so far as it is made by analytic philosophers, is one they make because they don't like Hegel. Most of them don't know any more about Hegel than I do, I'll wager. Personally, I'm a big fan of the Law of Non-contradiction, and I think there are good reasons not to reject it. But doing so, as I've indicated, isn't completely nuts. If you want to know about this approach to logic, read the article on dialetheism at the Stanford Encyclopedia.