# Logic textbooks which offer a system of natural deduction containing a so called "rule of replacement" restrict this rule to logically equivalent formulae. Only these can replace each other wherever they occur. I have often wondered why this is so. It seems to me that, having e.g. p q and p&r as lines in a proof (as premisses, say), would allow one to soundly infer q&r directly from them by replacement of p by q in p&r, without requiring that p and r be logically equivalent. In less formal situations, for example, when solving a math problem, I find myself (and others) doing this all the time. I've searched the internet for this, but couldn't find any answer so far. Most grateful in advance for a reply.

The question of which supplementary rules to add officially to a logical system as "derived rules", over and above the introduction and elimination rules for the connectives, is largely a matter of taste, of how you weight trade-offs e.g. between ease of use and spartan elegance. There's a good review of the choices made in various texts by Jeff Pelletier here .

# Omnivores are often defined as opportunistic feeders, in other words; they eat what they can get their hands on. As vegetarian sources of food are generally plentiful in the developed world; are there any valid reasons for eating meat? I’m finding it extremely difficult to think of any rational reasons for eating meat in my own life since I’m entirely able to survive on vegetarian options whilst still getting the nutrients I require. The strongest ‘weak’ argument I’ve came up with is it is ‘natural’ for us to eat meat – our bodies are able, and ready, to digest it. Like I said; this argument doesn’t win me over; there are many ‘natural’ things in this world that aren’t necessary for one to live a good life (and many more to contradict living one). For example; cancer is entirely natural – it is observed in the natural world. Likewise; the process of rape as a means of propagating has been observed in the animal kingdom (i.e. in chimpanzees and even dolphins), but I would never use the ‘natural’ argument...

Suppose someone asks: "What rational arguments can be used to validate drinking wine?" You can survive without wine whilst still getting the nutrients you require (well, so they tell me). But so what? Wine is a great pleasure to the palate, it makes you feel deliciously intoxicated, it is a delight to share with family and friends. ("Wine is sure proof that God loves us and wants us to be happy", Benjamin Franklin.) What better reason for drinking the stuff? Well, maybe you don't actually like good wine (shame on you!). But assuming you do, what more "validation" do you need? Likewise, let's sit down to (say) a wonderful plate of salami, prosciutto, coppa and lardo from cinta Senese, followed by perhaps ravioli stuffed with pigeon, then a tagliatta from Val di Chiana beef ... Well, food doesn't get much better than that: it is a pleasure to the palate, it makes you feel content and deliciously replete, it is a delight to share with family and friends. What better reason for eating the stuff...

# Look at this inference: Premise 1: All desks have the same color. Premise 2: That desk is brown. Conclusion: All desks are brown. Now, I understand that this is a deduction. However, the conclusion is a generalization of one of the premises, and generalizations of premises are what one would expect in induction. Where did I go wrong?

True: In any situation in which both premisses are true, the conclusion has to be true too. So the displayed inference is deductively valid. [There are possible wrinkles here, but let's ignore them.] Also true: Inferring the conclusion from the second proposition alone would be be an inductive inference, and a very bad one at that. The first is a fact about the given two-premiss inference; the second is a fact about a different one-premiss inference. So there is no conflict there!

# Many great thinkers are pessimists and often reach the conclusion that everything is pointless. Tolstoy even said that life is just a "sick joke". I started to read a lot of philosophy and I reach the same conclusion, that there is no absolute meaning and life is pretty pointless. And please don't reply that we should live in the now or we make our own happiness, etc.

The implication in the question, that Tolstoy was straightforwardly among the pessimists and thought that life is a sick joke, should perhaps not be let pass without comment. In A Confession , Tolstoy looks back at the period of his greatest worldly success. War and Peace and Anna Karenina had been received with immense acclaim. "I was not yet fifty, I had a kind, loving and beloved wife, lovely children, and a large estate that was growing and expanding with no effort on my part. I was respected by relatives and friends far more than ever before. I was praised by strangers and could consider myself a celebrity without deceiving myself." Yet despite all that, as Tolstoy eloquently reports, he found himself at a loss to find meaning in it all, and "gave up taking a rifle with me on hunting trips so as not to be tempted to end my life in such an all too easy function". In sum -- and I imagine that this is the passage that the questioner is alluding to -- Tolstoy writes "This spiritual...

# I have a daughter that is 14 years young. As a mother I understand that teenagers in her age grow up and they want to have fun, most of them with the guys. But still I can't let her go out. I think it's wrong. But my question is, Is that really wrong? Because I remember myself in her age... I also see the friends around her, they don't go out... well she's the only one. But she suffers because of me not letting her to have a boy-friend. Do you think I should let her? Because I'm really confused...

Just three quick afterthoughts, to add to Nicholas Smith's and Jyl Gentzler's wise but perhaps slightly daunting words. First, remember most teenagers do survive just fine (with a bit of a close shave here, and an emotional storm or two there): it is our burden as parents to worry far too much. So when your daughter tells you to lighten up, she's probably exactly right! Second, in any case, the big things that matter -- like your daughter's level of self-esteem, her self-confidence, how she regards men, and so on -- were shaped years ago. It's too late to do very much about them, and being over-protective won't help one bit. So the best thing you can do now is to be positive and supportive in her next phase of growing up. And third, to get back to the question originally asked: is it wrong to let her go out? Well, how could it possibly be wrong, if she's an ordinary girl wanting to do ordinary things? I can't see any compelling moral principle that has that implication. So just set some...

# I'm a mathematician looking at some of the work of Leonhard Euler on the "pentagonal number theorem". My question is about how we can know some statement is true. Euler had found this theorem in the early 1740s, and said things like "I believed I have concluded it by a legitimate induction, but at the same time I haven't been able to find a demonstration" (my translation), and that it is "true even without being demonstrated" (vraies sans etre demontrees). This got me thinking that "knowing" something is not really a mathematical question. A proof lets us know a statement is true because we can work through the proof. But a mathematical statement is true whether we know it or not, and if you tell me you know that a statement is true, and then in fact someone later proves it, I can't show mathematically that you didn't know it all along. This isn't something I have thought about much before, and my question is are there any papers or books that give some ideas about this that would be approachable by...

Perhaps there are two different questions here. There's a very general question about truth and proof; and there's a much more specific question about the sort of case exemplified by Euler, where a mathematician claims to know (or at least have good grounds for) a proposition even in the absence of a demonstrative proof. Let's take the specific question first, using a different and perhaps more familiar example. We don't know how to prove Goldbach's conjecture that every even number greater than two is the sum of two primes. Yet most mathematicians are pretty confident in its truth. Why? Well, it has been computer-verified for numbers up to the order of 10 16 . But so what? After all, there are other well-known cases where a property holds of numbers up to some much greater bound but then fails. [For example, the logarithmic integral function li ( n ) over-estimates the number of primes below n but eventually under-estimates, then over-estimates again, flipping back and forth, with...

# Over a year ago, I read Quine's Two Dogmas for a philosophy class. One part in it makes the step from talking about meanings to abolishing meanings and talking only about synonymy. I never quite got that. I mean, if there are two things similar (or the same) about something, don't they each have to have those things? If two pieces of string have the same length, they have each have a length, and they happen to be the same. Likewise for any other properties I could think of, such as color, volume, mass, etc. I don't see how sameness could not imply those "intermediary entities" which are the same. Thanks.

Consider an example from Frege: the direction of the line L is identical to the direction of the line M if and only if L is parallel to M. That's true. But how should we read it? Do we read it as explaining the notion of being parallel in terms of the identity of two abstract objects, i.e. two directions? Or do we take it the other way about, as partially explaining talk about two abstract objects, directions, in terms of the already-understood notion of lines being parallel? There's lots to be said for taking it the second way, as introducing reference to certain abstract objects in terms of something more familiar. Likewise: the meaning of "gorse" is identical to the meaning of "furze" if and only if "gorse" and "furze" are synonymous. That looks true too. But how should we read it? Do we read it as explaining the notion to synonymy in terms of the identity of two abstract objects, meanings? Or do we take it the other way about, as (hopefully) partially explaining talk about two...

# What happens to the souls of people who are in a coma?

Short answer: People (in comas or otherwise) don't have souls, so the question doesn't arise. Longer answer: The idea of a soul is, in one main tradition, the idea of an entity, quite distinct from our physical body, which can at least in principle survive independently of the body (and is often thought to be immortal), is the locus of conscious mental activity and is the initiator of our actions as self-aware agents. So, this idea of a soul goes with a dualist or two-component view of the person as compromising a material body and an immaterial soul or mind. Most contemporary philosophers of mind think that there are no good reasons to accept this kind of dualism, and very good reasons not to do so. For some of the arguments, see the opening chapters of The Philosophy of Mind: an Introduction , by myself and O.R. Jones, or any one of a couple of dozen other introductory books on the mind. So most philosophers hold that people don't have souls (that isn't, of course, to deny that...

# How does one go about becoming a philosopher?

And let me add a link to some reflections on a related question, on getting started in philosophy.