Logicians have developed many different theories of how logical arguments work, and the theories have plainly changed over time. Nevertheless, many standard examples of argumentation that logicians have long regarded as valid have been remarkably stable throughout history. As a result, what has changed is not so much the underlying reasoning that logicians have sought to capture as the ways in which they have tried to capture it. Specifically, when logicians think about deduction, they seek to capture the form of arguments, but logical form can be represented in different ways. For example, Aristotle sought to capture the reasoning behind classification--meaning arguments in which one predicate includes or excludes another predicate (or the ways in which one class includes or excludes another class). This became his theory of the syllogism. About a century later, by contrast, other logicians sought to capture the ways in which the truth or falsity of whole propositions could entail the truth or...

The human brain is a physical object, and many people think that logical relationships are the way they are only because our brains happen to work in a particular manner. But there’s a problem with this theory: Our brains work properly in the first place only because they recognize logical relationships—if they didn’t, we would reason incorrectly and would harvest at the wrong times, or drive in the wrong direction, or fail in our efforts to operate a computer. In that case, however, our brains’ mechanisms can’t define what counts as logic. Instead, logic helps to define what counts as a functioning brain. Since we survive only by making logically correct inferences, a correct sense of logical relationships is already part of what constitutes a viable brain, and thus it is the demands of logic, over the course of human evolution, that have shaped the physical structure of the brain, not the other way around. Another common theory is that logical relations are the way they are only in virtue of the rules...

Perhaps I can play the devil's advocate and rebuild the case for thinking that systematic ethical theory gets us nowhere. There are actually many different systematic theories--utilitarian, contractarian, deontological, etc.--but the trouble is they clash. The defenders of such theories often agree on particular moral judgments, but as to the abstract principles that define these systems, the experts disagree. In fact, it is precisely disagreement over the principles of these systems that animates much current academic debate in ethics. Yet if not even the experts can agree on which of their systematic principles are correct and which incorrect, why should anyone else rely on them? The theories in question are just as disputable as any real moral decision they could be invoked to justify. Again, systematic ethical theories are often defended on the grounds that they are like systematic theories in empirical science. (Rawls, for example, makes this move.) Yet empirical theories in science are reliable...

Some folks are enamored of the idea of “shooting dead an intruder in one’s house without asking questions,” but I’m not one of them. Necessary self-defense has always been a basic part of common law, but it is quite another thing to kill a man or woman on the sole pretext that that person is an intruder, whether or not the person constitutes a genuine threat to anybody in particular. Even jurisdictions that have stand-your-ground laws, or “castle” laws, generally require that you have a genuine fear of harm, or at least a genuine fear of a serious felony, before you are justified in using deadly force. And even then, the level of force must be necessary. Stephen Maitzen is entirely right when he points out that an undocumented alien is almost certainly not a genuine and immediate threat to you. Beyond this point, however, the notion that you can lawfully kill someone just because that person has intruded into your home now circulates broadly, and it is, in fact, legally false. You have no such legal...

Justice Joseph Story of the U.S. Supreme Court once wrote, “Every person accused as a criminal has a right to be tried according to the law of the land—the fixed law of the land, and not by the law as a jury may understand it, or choose, from wantonness or ignorance of accidental mistake, to interpret it.” Justice Story’s outlook has largely prevailed in American courts, and the standard formulation in most jurisdictions today is that though juries do indeed have a very real power to disregard (or “nullify”) a judge’s instructions in reaching a verdict, they are nevertheless duty-bound, morally, to follow those instructions, even though they cannot be punished for violating that duty. The task of a trained judiciary, according to this view, is to interpret complicated laws correctly—something that ordinary citizens are sometimes unequipped to do. The judiciary’s role also goes beyond the mere enforcement of procedures. Instead, judges must interpret complex statutes in the light of case law, and an...

Perhaps I could add something here too—and perhaps it will be useful: You are right that there is a difference between the two statements that you offer, and the difference has become more significant with the rise of many-valued logics in the 20th and 21st centuries. If one says, “A is either true or false,” then there are only two possible values that A can have—true or false. But if one says, “either A or not-A is true,” then there might be all sorts of values that A could have: true, false, indeterminate, probably true, slightly true, kind of true, true in Euclidean but not Riemannian geometry, and so on. The first formulation allows only one alternative to “true” (namely, “false”), but the second formulation allows many alternatives. The second formulation does indeed require that at least A or not-A be true, but it puts no further restrictions on what other values might substitute for “true.” (For example, perhaps A is true, and yet not-A is merely indeterminate.) The advantage of sticking to...

Your ability to balance your checkbook, or to draw logical inferences in everyday life, won’t be affected in the least by difficulties in figuring out just how logic and higher mathematics are connected. Nevertheless, the relationship between logic and mathematics has been an intriguing conundrum for the better part of two centuries. There have been many attempts to understand various aspects of logic mathematically, and perhaps the most famous is George Boole’s Mathematical Analysis of Logic (1847), which laid the foundation for Boolean algebra. Far from being a failure, Boole’s effort seems to have been a smashing success, especially when we consider the extent to which Boolean algebra underlies modern digital computing. Nevertheless, the relationship between logic and mathematics can go in two directions, not just one, and so, just as one might try to understand various parts of logic mathematically, one can also try to understand various parts of mathematics logically. It is this further...

Allen Stairs is right in suggesting that it is possible to lie by saying nothing, and perhaps it is worth adding that lies of this sort often form the basis of prosecutions for fraud. Fraud is legally complicated, but the basic idea is that you can injure someone by causing that person to rely on your assurances, even when you know that your assurances are false. Many legal systems assume that such conduct should sometimes be punished, depending on the circumstances, but they also assume that false assurances (or false representations) can come from what you don't say as much as from what you do say. For example, if I sell you a car without telling you that it has no brakes, and I omit this fact knowingly, my conduct might well be punishable as fraud. One can label such conduct with a variety of names, of course, but the basic idea is that there is something fundamentally wrong and dishonest about it, and many societies have sought to punish it. A key element in these situations is trust. If I sell...