# Notation: Q : formal system (logical & nonlogical axioms, etc.) of Robinson's arithmetic; wff : well formed formula; |- : proves. G1IT is always stated in the form: If Q is consistent then exists wff x: ¬(Q |- x) & ¬(Q |- ¬x) but we cannot prove it within Q (simply because there is no deduction rule to say "Q doesn't prove" (there is only modus ponens and generalization)), so it's incomplete statement, I don't see WHERE (in which formal system) IS IT STATED. (Math logic is a formal system too.) In my opinion, some correct answer is to state the theorem within a copy of Q: Q |- Con(O) |- exists x ((x is wff of O) & ¬(O |- x) & ¬(O |- ¬x)) where O is a copy of Q inside Q, e.g. ¬(O |- x) is an arithmetic formula of Q, Con(O) means contradiction isn't provable...such formulas can be constructed (see Godel's proof). But I'm confused because I haven't found such statement (or explanation) anywhere. Thank You Very Much

One thing the questioner seems to want to know is in what kinds of theories the first incompleteness theorem can itself be proved. As Peter says, the proof of the theorem is fine given informally, as almost all actual mathematical results are. But still, one might want to know: Where is this theorem provable? The answer is: Not in Q, but in fairly weak theories. It can certainly be proven in PA---full Peano arithmetic---though that hardly counts as a "weak" theory. I am fairly certain, though, that it can be proven in the theory known as "I-Sigma-1"---though I'd have to check that before betting my life on it. Part of the reason I'm sure about this, though, is that I-Sigma-1 is susceptible to the second incompleteness theorem. What you get in such cases is (e.g.) that PA |- Con(PA) --> G PA ; but then, if PA |- Con(PA), then PA |- G PA , and PA is inconsistent, by the first incompleteness theorem, and this works for any theory T that extends Q and is capable of proving: Con(T) --> G T . ...

# I studied philosophy in university and I recall that one of my tutors for symbolic logic was trying to walk me through a problem by saying that if you have a large enough set of premises, two of them will inevitably contradict one another. I've always had trouble understanding (and consequently, accepting) this proposition because: if one conceives of reality as a set of claims (e.g., I am right-handed, electron X is in position Y, 2 + 2 = 4, etc.) there are an infinite number of "premises" to the "argument" that is reality and consequently reality is self-contradictory. Am I missing something here? Can you explain which of us is right about this and in which sense? I should mention that I don't necessarily have a problem with reality being self-contradictory, but that really throws symbolic logic out the window (and doesn't throw it out the window at the same time)! Thanks to all respondents for their time. -JAK

Better: There's a fairly simple proof that this is false. Just consider the theory consisting of the sentence letters p1, p2, .... This theory is clearly consistent. It'd be an amusing question just how easy the proof is, i.e., exactly what sorts of theories are needed for the proof.

# Does the law of bivalence demand that a proposition IS either true or false today? What if the truth or falsity of this proposition is a correspondence to a future event that has yet to occur?

I take it that by "bivalence", you mean the principle that every proposition is either true or false. And if we take that principle in unrestricted form---we really do mean every proposition---then, well, it's hard to see how it could fail to imply that the proposition expressed by "There will be a riot in London on 13 January 2076" is either true or false. If you don't like that conclusion, then you have to abandon bivalence---or, perhaps, the claim that the sentence in question expresses a proposition, though that seems rather worse. But note that you do not have to abandon bivalence, so to speak, across the board. You might still think that every mathematical proposition is either true or false, or that every proposition about the past is either true or false, or.... Perhaps there is something special about the future here. As you probably know, Michael Dummett argued that one way to understand debates over "realism" takes them to turn upon our attitude towards bivalence regarding...

# How do we know if we are reasoning correctly? Consider, for example, this witty “proof” that a ham sandwich is better than eternal bliss: Nothing is better than eternal bliss. But, surely, a ham sandwich is better than nothing (despite Leviticus 11:7). Therefore, a ham sandwich is better than eternal bliss! Admittedly, the error in *this* argument may be easy to see. But, of course, in more subtle lines of reasoning it is much harder to check for bugs. How, then, can we be confident, in general, that our arguments are fallacy-free?

I've generally become pretty good at detecting fallacious reasoning, both my own and that of others. I'm not perfect, to be sure, but pretty good. So, overall, it seems that, in any given case, my chances of having committed an undetected fallacy are fairly small. How small depends upon the complexity of the reasoning. If I'm trying to prove a complicated theorem, I may have to go over the proof repeatedly, spell out all the details, etc, before I can convince myself that the proof is correct. So maybe in those cases, at least initially, the chances I'd committed a fallacy are higher than they would be otherwise. But that seems all right. It just means that, to be justifiably sure that I've not committed a fallacy in any given case, I may have to do different things: a lot in some cases, maybe nothing in others. But if I have done those things and come to the belief that my reasoning is correct, then I see no particular reason the belief should not count as knowledge. Of course, one can push...

# Dialetheist: "Some contradictions are true." My question: "Who claims (if any), that some tautologies are false?"

A dialethist would also make this claim. If p is a true contracdition, then it is both true and false. Hence it's negation is both true and false, but it's negation is a tautology.

# Alex George wrote [http://www.amherst.edu/askphilosophers/question/1663] that we can't ask "why should we be convinced by logic" or some similar question without thereby already submitting to logical priority; i.e., because the question itself has logic embedded in it. I'm not sure I understand this claim fully. Logic studies entailment relationships; if p, then q, therefore if not q, not p. On the other hand, logic doesn't tell us how to love another person. Insight from experience might tell us that. So there are other ways of knowing things, and different sorts of things, than logic. So if someone asks why choose to listen to logic at all, when I can learn plenty of important things from other roads to knowledge, why isn't this a fair question that doesn't already involve logic?

Recent epistemology has made a lot of the distinction between justification and something else that goes by the name warrant or entitlement , though some philosophers use "warrant" as an umbrella notion that covers both justification and entitlement. And of course the distinction gets drawn in different ways. But the basic idea is that being justified in some sense involves being able to appreciate that justification, which will, in central cases, take the form of an argument: If my belief that p is justified, then what justifies it must be something I know or at least could know. In that sense, justification, as philosophers in the tradition I'm exploring use the term, is more an "internalist" notion. An agent's being entitled to a belief, on the other hand, need not involve h'er appreciating the nature of that entitlement: I could be entitled to a belief that p even though I have no idea what it is that entitles me to that belief. And so entitlement is more of an "externalist"...

# Hi, I was thinking about the "This statement is false" paradoxon and so I came to: What about the "This statement is paradox" ? It means that I, the statement, can't be true or false. I find that odd. ..Jumping (1) Layer of statements: "I drink coffee" (2) Layer of statements about statements: " is true/false" (3) Layer of statements about statements about statements: " is paradox/not paradox" or is it: " is true/false-determinable/finite or not" Statements of (1) can state every possibility of language. Statements of (2) state if statements of (1) correspond with reality/each other. Statements of (3) state if statements of (2) are self-referential? finite? Where are my mistakes :p? Or which books do you advise me to read? Err..Which question should i ask? Does (3) "exist"? Is the idea of layers a bad idea? Simon

The idea that there is a hierarchy of statements, each saying something about the level below, but none of the lower ones saying anything about the higher ones, is central to formal work on truth. It originates where such work originates, with Alfred Tarski's great paper "The Concept of Truth in Formalized Languages". In Tarski's work, it doesn't take quite the form you mention. You suggestion regarding (3), in particular, sounds more like the form the hierarchy takes in Saul Kripke's treatment in "Outline of a Theory of Truth". On such treatments, the hierarchy certainly does not end with (3): There are also statements about statements about statements about statements, and so forth; and the hierarchy does not even end with all the finite levels.

# Quine's Paradox (“yields falsehood when preceded by its quotation” yields falsehood when preceded by its quotation) doesn't seem to me to be a paradox. Maybe I'm wrong, but it seems to me like it's asserting nothing but the fact that it's false. For something to be true OR false, there must be some other claim made. When I look at the statement, it seems to me that it's not talking about anything but itself -- like an indirect self-reference. It seems to me to have no content but its own claim that it yields to falsehood, and would therefore neither be true nor false. Have I made a mistake in my reasoning/logic?

No, I don't see any mistake---other than that you dismiss the problem simply on the ground that there is self-reference. Self-reference isn't always a problem. In fact, some times it's essential. Consider this phrase: (*) yields a sentence when preceded by its own quotation This is a perfectly sensible verb phrase, and some phrases yield a sentence when preceded by their own quotations---e.g., "is a sentence" does so---whereas some others---e.g., "Bill is"---do not. As it happens, (*) too does so. That is, (**) "yields a sentence when preceded by its own quotation" yields a sentence when preceded by its own quotation There's self-reference there, too, but what (**) says is true, and I'm not sure how else you'd propose to report that truth. Or should we just not report it? or try very hard not to think about it? or what? Now, among the phrases that do yield a sentence when preceded by their own quotation, some yield true sentences---e.g., "contains three words" does---and some do...

# I realize that this isn’t exactly a philosophical question but I’ve tried asking it elsewhere, to no avail. Can anyone help? I'm studying at home with two books on critical reasoning that are recommended in some philosophy introductions. I think I've done ok with most of the exercises so far, but I've really struggled with the sections on identifying implicit assumptions (context, underlying, additional reasons/ enthymemes, intermediate conclusions etc.). Is there anyone on this panel who is aware of any other resource which gives further opportunity to practice identifying implicit assumptions, and gives answers? Old law-school admission tests or something? I've gone through all the exercises in both books now, and unfortunately the answers are heavily etched into memory, so they’re no longer practice. If not then perhaps some general tips? Any help would be valuable, as it would seem to be a serious hurdle for me. I’m trying to ease back into the realms of academia after having three children and...

Unfortunately, I think the only concrete thing I can suggest is that you look for other texts on critical reasoning. (You don't say which two you have, and I'm not familiar with many, so I'm afraid I can't give more specific advice.)

# I believe that rich people are responsible for a lot of the problems in this society. In fact, I believe the rich often contradict their own interest. One everyday example, a new sexy automobile comes out and is in limited quantity the first year. Dealers put a premium on the car because of the limited supply. The rich pay the premium and buy the car. But if they refused to pay a premium, along with everyone else, they could get the car without paying the premium because the dealer would have to remove the premium. Therefore the rich have paid the premium against their own interest. Why are the rich so stupid and how do stupid people get to be so rich?

In the example you give, it seems obvious that there is another interest people have of which you are not taking account, namely, the desire to be "the first on the block" to have the new toy. I suppose one could question whether it is rational to have such an interest, but lots of people clearly do have it, rich or otherwise.