# Is there a difference between a number as an abstract object and as a metric unit used to measure things?

I would put the question slightly differently, if I understand it right: The question is whether the cardinal number 3, used to say how many of something there are, is the same or different from the real number 3, which is used to report the results of measurement. There is of course a different between the cardinal number 3 and a length of three meters, but the question is whether, when one says, "There are three apples" and "This board is three meters long", we refer to the same number three both times. Mathematicians and people who work on foundations of mathematics tend to have different views about this, at least in practice. The way one defines the cardinal numbers in set theory, for example, is very different from how one defines the reals. But working mathematicians will often speak of "identifying" the cardinal with the real and often seem impatient with such niceties as whether they are really the same. A more difficult question, I think, concerns cardinals and ordinals....

# Has there been much work done on the notion of approximate truth, for example under what rules of inference approximate truth is preserved, or what kind of metric one could use to say that proposition X1 is 'truer than' proposition X2?

Yes, there's quite a good deal of work on this kind of thing. It tends to go under the name "fuzzy logic" or "degree theoretic logic". The Stanford Encyclopedia entry is a good place to start. There's a fairly recent paper by Brian Weatherson called "True, Truer, Truest", if I remember right, that does some work on the philosophical foundations, which have always been a bit unclear.

# I have a question about existence withing a formal system. Can we construct it so that a theorem t implies "there exists" theorem t itself? Thanks, Paul

I'm not quite sure what the question is here, but here's what I think is meant: Can we construct a statement S such that S implies that S itself exists? If that is the question, then the answer is "Yes", assuming we have some fairly minimal syntactic resources (namely, those sufficient for the purposes of Gödel's theorem). If we have those resources, then we know, by the so-called diagonal lemma, that, for any formula A(x) we can find a sentence G such that the following is provable: A(*G*) <--> G where *G* is itself a name of the sentence G. G itself will be fairly long, but is something that can actually be written down. So now let A(x) be the formula: ∃y(y = x) I.e., this means "x exists". Then, by the diagonal lemma, we have a formula G such that we can prove: ∃y(y = *G*) <--> G I.e.: G itself is provably equivalent to the statement that G exists. We can in fact say more. If we have the syntax we need, then we can prove that G exists. Hence, we can prove G itself,...

# Does "if p=q, then nec(p=q)" hold, if "p" and "q" are intended to denote properties? I am told that it holds. But it doesn't seem to be quite right. It seems to depend on what it is for two properties to be identical. Am I confused?

If two properties P and Q are the very same property , then how could they have been different? One could, indeed, have discussions about what it is for P and Q to be the very same property. But, if true identities are always necessary, then that fact acts as a constraint on theories of property identity. For example, suppose someone suggested that properties are the same just in case they are (actually) true of the same things. Then it would be an objection to this view that there are properties that are true of the same thing but might not have been.

# I aced a basic logic class in college that covered both sentential and predicate logic. I am interested in furthering my skills in symbolic logic, but I don't know how. My school doesn't offer any upper-level logic courses. I'm thinking I would like to buy a simple textbook for a more in-depth study of the more advanced concepts (I've heard the term "modal logic" thrown around, but I don't know what that is). Can you suggest a good text or author I should investigate?

Peter might also have mentioned his book, An Introduction to Gödel's Theorems , and the similarly targeted book by George Boolos, John Burgess, and Richard Jeffrey, Computability and Logic . Both are standard texts used in intermediate logic courses.

# Is it true that all people are beautiful? Or is that just a white lie we tell to make non-beautiful people feel better?

Just to be contrarian, let me perhaps disagree with what my colleagues have said. It seems likely that the term "beautiful" is what philosophers call "context-sensitive". That is, what it means varies from case to case. The simplest examples of such words are "I", "you", "here", and the like, but most of us have come to the conclusion in recent years that most expressions of natural language exhibit some degree of context-sensitivity. For example, quantifier words, like "all", seem to do so. The sentence "Everyone is on the bus" certainly need not mean that absolutely everyone is on the one and only bus in the universe. What it means clearly varies from case to case. The same seems to go with "beautiful", and in two respects. One is that "beautiful" is a scalar adjective, like "tall", in that it accepts modifiers like "very". And, like "tall", how beautiful something has to be to count as beautiful tout court will vary from case to case. That might make it possible truly to say "Everyone is...

# So, it's my understanding that Russell and Whitehead's project of logicism in the Principia Mathematica didn't work out. I understand that two reasons for this are (1) that some of their axioms don't seem to be derivable from pure logic and (2) Gödel's incompleteness theorems. However, particularly since symbolic logic and the philosophy of mathematics are not my area, it's hard for me to see how 1 & 2 work and defeat the project.

I think it's important to distinguish the two sources of "failure", not so much as regards Principia but as regards logicism quite generally. I'll stick, as Prof Smith did, to arithmetic. Here's a way to put Gödel's (first) incompleteness theorem: the set of truths expressible in the language of first-order arithmetic cannot be listed by any algorithmic method, i.e., it is not (as we say) "recursively enumerable". Now why is that supposed to show that logicism fails? Because the set of theorems of any first-order formal theory is recursively enumerable. This is a consequence of Gödel's first great theorem, the completeness theorem for first-order logic (and also of what we mean by a "formal" theory). So the truths aren't r.e. and the theorems are—you can list the theorems but not the truths—so the theorems can't exhaust the truths. Now why is this a problem for logicism? Obviously, as the argument has been stated, it depends critically upon the assumption that the proposed logical...

# Why is it that when a white person says a racial slur, such as "nigger" it is thought to be the most heinous crime. However, when a non-white, in particular blacks call whites "crackers" it is dismissed as nothing. Why is there such a double standard in American society? Why is reverse racism rampant more than ever? Whites have to fear of being shunned for voicing their injustices, because if they do, they will be called a racist. If a white is mistreated due to race in the work place nothing occurs. On the other hand, if it happens to a black it gets mass media coverage. The politics are backwards, the NAACP, pushes racial equality for blacks, yet they are immersed with racism towards whites; not all are but it has been displayed. If a white were to make an Organization for the advancement of their race it would be an outcry for its dismantle. Shouldn't all race Organizations be abolished since we're under the same umbrella, the Human race? I too often experienced this firsthand, being of black decent. I...

The questioner makes a number of factual claims which seem to me to need rather a lot of support. In fact, I'm not sure that any of the factual claims the questioner makes are correct. Who is it that dismisses racially charged remarks by blacks as "nothing"? What examples of workplace mistreatment due to whiteness does the questioner have in mind? Which of the NAACP's leaders are racially biased, and what is the evidence of that bias? Where is the evidence that "reverse racism" is rampant? Are whites being randomly stopped by black police when driving through black neighborhoods? Are whites suddenly more likely to receive jail time for drug crimes? or to receive the death penalty for capital crimes? Have dozens of studies shown that a job applicant whose details (e.g., name) make it clear that he is white is less likely to be interviewed than one who is clearly black, even if all relevant details of the CVs are otherwise identical? Have similar studies shown the same thing about...

# In ZFC the primitive "membership" usually has the statement "x is an element of the set y". My question is "is the element 'x'" of a set ever not a set within ZFC?

To add a bit more, there are some interesting applications of urelements in set theory. Perhaps the most famous example is Quine's theory New Foundations. NF, as it is known, which does not permit urelements, remains something of a puzzle: It is not known if it is consistent. But NFU, which is just NF plus urelements, is known to be consistent if Peano arithmetic is. See the wikipedia entry on NF for more. I seem to recall some interesting results due to Vann McGee about ZFU, as well, but they do not come immediately to mind.

# How do you show some conclusion of an argument cannot be derived in a complete system? Does one have to make the truth table to show that it is not valid and therefore, by definition it should be impossible for that conclusion to be derived?

This question has essentially been answered, I think, as question 3211 . In brief: A truth-table is one way to show that a conclusion cannot be derived, but it is not by definition that it cannot be. This is a consequence of the soundness theorem , which states that every derivable formula is valid. Since the truth-table shows the formula is not valid, then it cannot be derived since, otherwise, it would be valid. That said, a truth-table is not the only way to show that a formula cannot be derived. For one thing, there are also trees (or "tableaux"); for another, there are purely syntactic arguments that can be given to show, e.g., that a formula containing no more than one occurrence of any sentence letter cannot be derived. See question 3211 for a similar example.