Talk:Mathematical logic

494:"logic" actually is. And I am not sure what it is, exactly, as I usually need only first order. But I would imagine that "language = logic + signature", and that deduction rules are related to a logic almost like structures to a signature. I think making clear the modular character of these concepts should really help to get a uniform terminology that makes sense for people from various branches of logic and from universal algebra – necessary for weeding out duplication. (I am not saying this should be part of this article – I haven't thought about it. It's just something I thought I would do some time, and which might be relevant here, perhaps even at an early stage.) In any case, thanks for doing this. I am sure I am going to learn something from the final result as well as from the way you go about it. -- 2784:"Truth" in mathematics is a technical term. It is not universally accepted that a formal sentence can be "true" or "false" in an absolute sense at all - "truth" is a model-specific property. The fact is that there provably exist nonstandard models of arithmetic that obey the peano axioms, and whose theories do not contain contradictions. To claim otherwise involves importing a distinctly non-formal definition of "truth" and "consistency." It is true to say that the Goedel sentence is true *in the natural numbers*, but it's not well-justified to treat truth in the natural numbers as some sort of privileged "absolute truth." 4870:

mathematical logic on my personal bookshelf, there's some peripheral mention of modal logic, but nothing about the other four kinds of logic, and there's nothing about runtime code generation either. Maybe this table is related to some kinds of computer software packages for symbolic logic. Either way, it either needs a really good explanation (and citations), or it needs to be removed. If it's about computer logic software, maybe there's another article where this table will make sense. Mathematical logic is already difficult enough to understand without interpolating (at best) peripherally related material into it.

2731:

and expression of logic by means of typically mathematical disciplines (or in other words the application of mathematical theories and systems to logic). This is especially the case historically, with the developments brought about by the work of the likes of De Moivre and in particular Boole (whose algebra fit perfectly logic concepts that were earlier handle through different means). The history of the intertwined relationship between logic and mathematics, I agree, is far from linear and id still in fruitful progress, but for such reasons I endorse a two-way view on it.

5096:

view of the subject and one view of history, and some assertions seem to be false. Just adding citations of classic papers from a hundred years ago does not justify at all the picture which is painted of the scope of the whole subject and its history, nor of the particular assertions which are made. What is needed is more references to particular pages of particular reasonably recent books which cover this subject, especially for assertions which are dubious, controversial or tendentious. I definitely have the impression that this article is inadequately justified.

3190:

in the natural language sense than statements that are true in any other model. And I'm going to similarly "boldly assert" that I'm not the only person who thinks this way, and that it is bad for this encyclopedia to assume that everyone does. The wording that I suggested is non-contentious regardless of whether or not you think that the natural language meaning of "truth" somehow lands closer to one model than to another - it is best to assume as little as possible, and leave interpretations to the individual.

95: 85: 3553:(edit conflict) Well, if you really want to get into that way of thinking it might have to do with the published positions about the philosophy of mathematics, which are many and various. I don't see why Goedel's view should be given special weight - mathematical theorems belong to everybody, and Tarski wasn't a goof-bag either. But really we should just include all the details so that this isn't even an issue and everybody can understand it regardless of their philosophical dispositions. -- 64: 31: 294: 380: 330: 651:

definition of continuity. My impression from what I have seen on the web is that it was first used by Bolzano, then more rigorously by Cauchy (who actually made wrong claims because he didn't think of the problem of uniform convergence), and then rigorously by Weierstraß. The following article should have more precise and reliable information, but as usual I can't read it from home: Walter Felscher,

22: 725:. But they also include the model relation, which may not be needed for proof theory, and no inference rules. I was looking for a word just for a functor from signatures to languages, which could then be equipped with functorial model relations and functorial inference rules. — Yes, I imagined that you left the model theory bit for me. I am thinking about this. 429: 196: 169: 4189:

that some model of the group axioms is more special or "real" than another. The issue being taken here is not one of disputing foundations, it is disputing the use of one's own natural-language interpretation of a formal sentence to provide a distinctly non-mathematical interpretation of something which does not necessarily have any intrinsic meaning.

2216:. There are almost certainly some minor areas that will be hard to categorize as proof theory, model theory, recursion theory, or set theory. But most of the "general logic" category is considered proof theory in practice. On the other hand, not everything that involves the word "logic" and is studied by mathematicians is part of mathematical logic. 4334:(1) Do not take this occurrence of "God" literally. (2) No, we cannot do all that (nor even a large part of all that) just because "precise definition of concepts like languages and models, and the axioms of arithmetic and set theory that we are talking about" is much, much longer than "axiomatic definition of a group that everyone agrees with". 206: 2542:, consisted of the single sentence "The study of the meaning and relationships of statements used to represent precise mathematical ideas. Symbolic logic is also called formal logic." with a few "see also" pointers. It might be reasonable to merge some of the old content of "symbolic logic" that didn't make it into this article, into 2802:. (In particular, it is not about the objects of discourse of the formal theory whose Goedel sentence it is, unless that formal theory happens to be a theory of arithmetic. So for example the objects of discourse of ZFC are sets, not natural numbers, but the objects of discourse of the Goedel sentence of ZFC are natural numbers.) 882:

minds. His home page has a second draft from January 2003 and more corrections from November 2003. He also says there that he is working on a simplified example which he hopes to have complete "in June" (presumably June 2005, since the page was last changed in May 2005). So unfortunately it looks like this example is dead.

3189:

Actually, that's precisely what I mean when I say that. I do not think there is something fundamentally special about the natural numbers and that when I speak of arithmetic I am speaking with those in mind, or that statements that are true in that particular model are somehow closer to being "true"

2915:

Well, no, it really doesn't. If there are models, then that's enough realism to justify the claim that the Goedel sentence of a consistent theory is true. If you're enough of an anti-realist to have doubts about that proposition, then you owe an explanation about what you can possibly mean when you

2878:

Well, close. There isn't any "the" hyperfinite numbers, per se. There are various nonstandard models (but only one standard one, up to isomorphism). Some of the nonstandard models satisfy the sentence, and some don't. But the standard model definitely does satisfy it, and that's what "true" means

2829:

I agree with the IP editor that Knowledge articles should not take a platonist POV. Even when authors say it's "true" we all know enough to realize they may only mean that it follows from ZFC, and may or may not endorse the notion that ZFC is "real." Anyway, what is the problem with adding a few more

2730:

I have an objection to the restricted sense given to the expression "mathematical logic". It is true that on the one hand it means what you say, that is the exploration of mathematical concepts and workings by means of logic. On the other hand, in a way the converse is true, that is the formalisation

847:

I had never heard about this result before I read it in this article, but since it fell into a period in which I was not very active in mathematics I assumed I had just missed it and was going to read it. Now based on your warning I have asked an expert, who told me that there is in fact no consensus

3797:

Footnotes look like a solution. It is also proposed above. Many readers will head off faced with a formal definition of "true but not provable within the theory" in an early section. I dare say also that the mathematical formalist will not be confused at all, since it is extremely unlikely that this

3461:

I am not an expert in logic, but my feeling is that the POV you mention is in fact the mainstream. The "true but unprovable" statement about the machine is in fact proved in a meta-theory (of the given theory). Due to Goedel we know that a theory cannot be its own meta-theory. A nonstandard model of

3013:

If I'm reading this article these are the main things I want to know, not asides and footnotes. You can generally make math articles look nice by glossing over a lot of fine points, and you may feel like this makes things simpler but actually, to someone unfamiliar with the subject, it ends up being

2286:

It appears to me that "proof theory and constructive mathematics" are being lumped together here, since otherwise we'd use the Oxford comma. However I'm not sure it's an entirely defensible togetherlumping. While it is true that historically a lot of proof theorists have come from an ontologically

1919:, and I have not merged any of it anywhere. Some of it seemed to be wrong, and some of it is very elementary and doesn't really belong in such an article anyway. If someone wants to salvage something, there is a huge choice of articles related to propositional logic; perhaps one of them is suitable. 1914:

so it is easier to find. I think this takes care of JoesphPGrant's concern. A small number of languages still have the split, probably because they got it from us. The Finnish Knowledge has only an article on Symbolic logic, so I have added their interwiki link to the present page. There wasn't much

1079:

I reworked some edits to the "Early history" section. Stuff about the 19th century belongs in the following section. In order to have a global viewpoint, and avoid historical myopia, we do need to recognize that non-Western cultures had their own traditions of logic. The dominance of Greek influence

881:

So far as I know it was never published. There was a special session on this at the British Logic Colloquium 2002, so I would assume that's how it became well known to people outside stability theory. The example is extremely complicated, and so it probably took some time for people to make up their

742:

As it stands, the article now has a lot of information about historical developments, and some information about milestones in particular fields. I think it's weak on analysis, criticism, and other "secondary source" material. So my next goal is to try to add a little more criticism (preferably with

702:

I'm not sure about the citation tags. Mentioning the original year in prose is fine with me. One area where I am not strong is the early history of model theory, which is why it is currently just a single sentence about Tarski. You thoughts above about the nature of logic are relevant to the section

5283:

I have read this 10 times over and it still doesn't make sense to me. "Mathematical logic, also called formal logic, is a subfield of mathematics exploring the applications of formal logic to mathematics." This seems like an unnecessary circular definition. I haven't looked at the revision history

5153:

I think the previous section of the talk page was very apropos, and I have reverted that section of the article to a previous version, which is at least somewhat better. I agree with what you say about long lists, so why not simply mention the main two issues you see? I will attempt to improve them

5135:

I prefer not to make a long list because I don't want to get into prolonged discussions. Life is too short of unproductive adversarial arguments. (Editors on wikipedia typically defend their opinions to the death, especially if they are wrong.) One merely has to read through the article to see that

4581:

I would dispute that standard statements in mathematics are "realistic", but more importantly there is a significant distinction here, which is that the statement we are arguing about can cause confusion. All cases of "true but unprovable" amount to "provable with a certain set of rules, unprovable

4287:

they don't go and discuss ontological questions, but they do give an axiomatic definition of a group that everyone agrees with, instead of just saying, for example, "groups describe symmetry." In this article, we should have a precise definition of concepts like languages and models, and the axioms

4188:

a foundational choice about mathematics - we begin with axioms, not models). It is one thing to say "A group is a mathematical structure obeying these properties" without further qualification, and quite another to say "this sentence in the language of groups is 'true'" because you happen to think

3174:

be true, but since we can't try every case, we can only assert it's true if there's a proof. So you say a statement is true exactly when there is a proof that it is true, which comports with my original intuition that those are the same thing. Anyway, instead of presupposing that naive readers will

2994:

you need to parse out that a statement of arithmetic is "true in the standard model" just in case...now you do the whole Tarski thing inductively on the statement, and the way the "standard model" comes in is that you let the variables range over zero, one, two, etc, and interpret plus and times to

2987:

The problem with saying "true in the standard model" is that it makes it sound complicated, whereas it's really very simple. The reader has to ask "what's the standard model?". Oh, well, first you need to know what a model is. That's not deep, but it's a fair amount of structure to absorb. Then

1715:

be used sometimes as a synonym for basic proof theory. But I don't see any way to classify model theory and recursion theory as part of "the area of mathematics which studies the purely formal properties of strings of symbols." Now, I don't really think that is a good definition of "symbolic logic"

650:

for doing all this work. It's soon midnight for me, so I will probably print the article tomorrow morning to get a better overview. Believe it or not, I learned something very important about model theory from you today. — You noticed that we have contradictory information on the origins of the ε-δ

5095:

Although there are many inline citations in this article to classic historical papers, there are very few references which justify the picture which is painted in this article of both the history and current state of the subject. Many of the paragraphs seem quite dubious to me, just one particular

4644:

The case of groups, by the way, is a bad analogy. In group theory, we study groups, not group elements. The "axioms" of elementary group theory are axioms in a different sense — they're really part of the definition of group, which is the object of study. Arithmetic, on the other hand, does not

4409:

Do not think that I am a 100% platonist. I know that my idea of integers is imperfect (and no wonder; I am just a mortal). But I also know that my idea of integers is much more specific than expressed by the well-known formal languages. I definitely prefer models that make "Consis T" true (where T

3851:

Agreed, I don't find that "true in the natural numbers" matches my intuition for the natural language word "true" at all (and thus reading "the Goedel sentence is true" seems to me a terrible stretch of the word "true"), regardless of whether Goedel intended his theorem to talk about that specific

3778:

insofar as possible. I think it ought to be expanded a lot with a lot of explicit details, especially by making clear exactly what formal systems we are talking about. I have no objection to using language like "true but not provable," but only provided it is made abundantly clear what this means,

3508:

This is not political. Common sense says we should try to make the article intelligible to readers who may misunderstand the statement if they are formalists. (And formalism is not a "fringe" position, by the way - I'm not sure what the predominant view of mathematicians is, but I don't think it's

2757:

The Goedel sentence of a consistent theory is true. That is what reliable sources say, and is also the fact, and that is what we should say. Not to say it risks continuing the confusion promulgated by the popularizers, all this nonsense about the sentence having a truth value that is in some way

2010:

The MSC is not, of course, the controlling definition of "mathematical logic". The article here does mention algebraic logic and categorical logic, but I don't think they should be very heavily emphasized (algebraic logic should be covered in more depth than categorical logic, which should just be

1101:

We certainly need to have a global viewpoint. However, failing to note the Greek influence in Western cultures is also a mistake. There's a clear historical path from Aristotle, the Stoics, medievals, and through to the logical systems as developed and extended by logicians such as Frege, Peano,

863:

This is very interesting. I'm no model theorist, and people I respect referred to Knight's result as a proof at some point, so I assumed they were correct. Wasn't Knight's result published? Thanks for correcting my error, in any case. I'm glad other people are watching these pages. Are there other

738:

I agree the easy changes are getting harder to find, which is grear. There are still a few gaps in the coverage, and I made a list tonight of several more primary sources to cite, but I think the coverage is filling out well. This is good, because the article is approaching the recommended maximum

692:

An observation: The Citation tag has the year of the first edition as "origyear=1967", although I don't know if that's the correct use (since the current edition seems to be a reprint of the second edition from 1973, and I found no documentation on the intended purpose of origyear). When I started

4183:

One can assume a certain standard foundation for unqualified statements without believing that that choice of foundation is somehow more "privileged" or "fundamental" than another. Moreover, there is a stark difference between agreeing that unqualified statements about mathematical objects share

3369:

I think, the meaning of "true but unprovable" is easier to realize on the well-know example of a Turing machine that seeks a proof (in a given formal system) that it (this machine) will never stop and, if found, it stops. If the formal system proves only true statements, then clearly, the machine

2945:

Well, I'm not a realist about anything (not even about whether I'm a realist.) Maybe I ought to think about it more (or less.) I think that many readers won't get all this subtext so this seems like a perfect place to explain what is actually happening. A statement like "P is true but unprovable"

2233:

mathematical logic, or anything else. For example, the MSC has a whole section on computer science (68), but this obviously doesn't mean that computer science is claimed to be part of mathematics. Similarly, simply because some topic is classified under 03 does not mean it is really claimed to be

1599:

I would never have found my way here (Mathematical logic) without the separate Symbolic logic page, which I found very useful in itself, using terms familiar from my studies back in the 60's. There are authors (e.g. Suzanne K Langer, 1937) who view Mathematical logic as a sub-divison of Symbolic

1141:

Is it a core part of mathematical logic? No, certainly not. But it is important for the more philosophical side, and has the interesting applications to provability logic that are mentioned. So two sentences seems to me like a reasonable amount of time to spend on it, in the spirit of being "just

5172:

Good idea. I'll review the page and make a short list of comments over the next few days. I know enough to say that something is dubious, or even wrong. But I am not a specialist in this area. So I prefer to let others do the fixing, as you have done. Hopefully other people will contribute their

4639:

OK, I'm sorry, I find this weird; it's hard for me to buy that you're making this argument seriously. The natural numbers have been understood at least since Pythagoras (they left out 0 and maybe even 1 but that's a trivial difference); the axiomatizations are maybe 150 years old. It should be

4234:

Axioms for the (pre-existent!) integers are chosen by us mortals; no wonder that they are not so good as we hoped. This is why I doubt that "we begin with axioms, not models". Moreover, now we know that the Euclidean space applies to diverse problems, not only as an approximation to the physical

3494:(EC with Tsirel) Not all POV's are equal. This is not intended as rudeness, but merely stating that minority POV's need not be mentioned (or the majority POV be "explained") if the minority POV is fringe. It is the other way around. The minority POV needs substantial qualification and reference. 2029:

Not all of mathematical logic is formal, by the way. The meaning there is like the distinction between "formal proof" and "natural language proof". Fields such as set theory and model theory are usually conducted using natural-language proofs, rather than being explicitly treated within a formal

1822:

have meant, but it's not what it means in contemporary discourse. Mathematical logic is a collection of fields of mathematics that have some historical connection with logic — as Carl says, set theory, recursion theory, model theory, and proof theory (and one probably ought to throw in category

830:

I noticed that in the section "Model theory" it is written that Robin Knight refuted Vaught's Conjecture. However there was an error in Knight's 2002 construction and circa 2003-2004 there were attempts to patch it, but not everyone was satisfied with his arguments. Unless I've missed some new

512:

My first goal is to expand the depth of historical information and to describe the subfields in more detail. I have found it remarkably difficult to find reliable sources that speculate on the nature of logic itself, or define mathematical logic. This is likely because of the culture within math

5299:

I think you're right. Part of the solution is to remove "also called formal logic", which in addition to making the sentence circular, is also wrong. Mathematical logic is the discipline comprising computability theory, proof theory, model theory, and set theory (and I would tend to argue for

3718:

A lot of people have trouble understanding how a statement can be true but not true. A simple example, though NOT a Gödel sentence, is "This statement is not true." If it were false, it would state that it is true. But false statements are never true. Therefore it is true, and thus not true.

667:

Regarding the reference to Shoenfield, I think the inline citation should use the year of original publication, because this identifies the era in which the content was written. Later republications are important for purchasing the text but unless the context was changed they aren't going to be

4680:

Well. Rick's new language says "in the natural numbers", which you seem to have indicated would satisfy you, so hopefully we can move on. Actually even from my perspective I suppose it's useful to point out that it's in the natural numbers, to avoid confusion with the objects of discourse of

493:

Excellent. Unfortunately I am going to be rather busy in real life for a few days, so I can't help much before the end of the week. Just a thought, as I suspect you might be planning to go into rather more detail than there is right now: I don't know if we currently have a definition of what a

3157:

My intuitive notion of mathematical "truth" is more like the first thing you said, but I can imagine someone thinking along the lines of your second option. I'm not sure that everyone does, because they would then all say that the Riemann Hypothesis is true because if you apply Riemann's zeta

2073:

formal, that is mechanize proofs are rare in mathematical practice. But my understanding is that mathematical logic deals exactly with the metatheory/metalogic of those rather than of the natural language proofs, with the assumption that one can mechanically formalize them if necessary. (This

1991:

etc. I know this stuff isn't normally included in mathlogic textbooks (well, Peter B. Andrews's book cited here is an exception wrt to type theory), so no they should not have more than a passing mention here, but even that is currently lacking (except for type theory in the history section).

1125:

In computer science, modal logic is important because of the (relatively) good complexity classes of its various decision procedures. In philosophical logic it is important because of its more natural relationship to natural language. But are there any areas of core mathematical logic where

1010:

ought to be fixed; I left a note on its talk page. That outline confuses two things at present: topics that are learned when studying CS, and things that are part of the CS research landscape. The latter are what constitute "branches" of CS. If all the prerequisites counted as "branches" then

4869:

Yes, that's what I think too. But I'm not a top expert on this stuff. People who know more than me might know how this kind of thing fits in. So I'm reluctant to delete other people's contributions. I prefer to leave that to people who fully understand this topic area. But in the 45 books on

3908:(Edit conflict) I am afraid, this is impossible. About 50 years ago :-) I've read a book with full theory, hundreds of large pages... well, with proofs; but formulations could not be stated in less than 40 pages or so... definitely non-encyclopedic. You'll hear "try Wikibooks or Wikiversity". 2647:

Formal logic is usually divided into two parts, propositional calculus and predicate calculus. Because "calculus" has today a more common meaning: "the study of limits, derivatives, integrals, and infinite series", I prefer propositional logic and predicate logic, but both phrases are used.

3915:

This is the burden of being a formalist. A human explains to a human programmer in 15 min what is needed; and then the programmer explains the same to a computer during 3 months (now, when programming languages are very powerful; in 20 century it was 12 months). Why? Since the computer is a

1768:: "the formal mathematical study of the methods, structure, and validity of mathematical deduction and proof." Most philosophers would disagree that logic only deals with mathematical proofs. So, MathWorld is not necessarily the best source for definitions that aren't strictly mathematical. 559:) is largely independent from this one and twice as long. Its introduction makes some interesting points (alas, without footnotes). If you can't easily read French I can put a quick translation here. (And I really like the two footers they are using for the mathematics and logic portals.) -- 2954:

it's neither provable nor disprovable within the theory, and also by the way mathematicians may say something is "true" when they really mean that it is provable in the standard model. That's typically the right way to go on all these kinds of issues. More details leads to less philosophy.

3702:

A lot of people have trouble understanding how a statement can be true but unprovable. A simple example, though NOT a Gödel sentence, is "This statement is unprovable." If it were false, it would state that it is provable. But false statements are never provable. Therefore it is true.

2515:

is that it's simply an older term for mathematical logic (which by the way is not the same thing as "the logic of mathematics"). It's a term that carries a certain amount of philosophical baggage, that baggage being the most likely reason it's fallen into disuse. If you have a different

5000:

The idea that preventing self-referential propositions in first-order logic prevents Russell's paradox also seems wrong. You can very easily make minor changes to ZF and NBG axioms to permit Russell's paradox to occur. So I don't see that FOL has any power to stop the paradox at all.

3754:

theory, right? Gödel theorem (say, for arithmetic) is a theorem of meta-arithmetic, right? Thus, it is unspecified until you specify not only the formal arithmetic used, but also the formal meta-arithmetic used. Thus, our articles are anyway far not specific for a formalist, right?

2946:

sounds absurd, or worse, profound, to the uninitiated, and perhaps that's part of what fuels those popularizers you keep (justly) complaining about. You can use this as an opportunity to explain to the reader that the Goedel sentence of a consistent theory is true for the model

4504:

It is standard in mathematics to state results "realistically" even if you are not a realist. Saves a lot of time, for one thing. Perhaps more importantly in this case, you don't want to conflate an assertion itself with the provability of that assertion, even if you're a

1887:

shows that the articles were merged in the past, but then forked again just because a confused editor found a loopy link from the present article to itself via the redirect at Symbolic logic. Apparently nobody ever had a clear idea what the scope of that article should be.)

1584:

by Hamilton, and the (usually) informal logic used by mathematicians to prove theorems. Proofs of theorems in refereed journals almost never use formal mathematical logic, unless the topic of the paper is formal mathematical logic or, sometimes, axiomatics or set theory.

760:

Yes, the article seems to be converging very well. I am glad you have embarked on this experiment, which really looks like it's going to be a great success. I am not sure that I want to know how many hours you spent on it. Did you have time for eating during the weekend?

3438:

I'm all for an article called "true but unprovable" provided there are a few sources for it. However if you read the discussion between myself and Trovatore above it ought to be clear why this is a POV phrasing and should not appear in the article without qualification.

983:. The outline is right: Computer science has mathematical foundations, and these are in some sense considered to be part of computer science. But not everything that belongs in such an outline is a branch of computer science. Almost everything is in section J of the 3028:

I would argue that this is exactly the intuitive, natural-language meaning of "true", and should not be a surprise to the naive reader (except insofar as that reader expects all truths to be provable, but that's a surprise as to what happens, not as to what things

3636:

states that Gödel thought CH to be wrong. This is not typical of a formalist, but rather of a Platonist? Believing the CH wrong (or right) and at the same time accepting competing theories for the natural numbers isn't very consistent. I don't think he did that.

2998:

These are the kinds of things that can be dealt with in asides; explanatory footnotes (as opposed to references) are especially good for this. I think that would be a good solution here. But there shouldn't be any weaseling about whether the statement is true.

3204:

To both of you: I'm sorry, but I don't believe you. I don't think that's what you mean at all. I think that's a story you tell yourself to avoid coming to terms with questions involved with real existence of abstract objects. But it's not a natural reading.

3014:

completely opaque. Particularly since this statement involves a specific notion of "true" which may be in common use among mathematicians but is not commonly understood by the article's target audience (e.g. ignorant people like me), that should be explained. --

2805:

It is important, and sourceable, to say that the Goedel sentence of a consistent theory is true. There is far too much confusion on this point, especially from the popularizers, who like to give the Goedel results a sort of misty glow of something that can be

3139:

I don't know you, but I'm going to boldly assert that no, that's not what you mean at all. What you mean is that if you take any natural number, say 4 or 17 or Graham's number or whatever it be, there really are four natural numbers whose squares sum to that

4282:

More importantly though, I think you misunderstand what I am asking for. I don't want to hash out the whole realist v. formalist debate on every math page; rather, I just want explicit definitions made available so that it becomes a non-issue. In the article

4699:

Yes, the current wording seems to me perfectly satisfactory, so I think the current discussion has run it's course - we don't need to generate absolute agreement on the nature of mathematical truth here, only arrive at an acceptable wording for the article.

2562:

It looks to me like the old symbolic logic article was pretty scattered and the topics in it that aren't in the mathematical logic article, are still to be found elsewhere. So I'm thinking of removing the "missing treatment" tag. Let me know your views.

2048:

A classification system by necessity appears to give equal importance to the topics it includes. (Well, except for their relative placement in the tree). Of course, we shouldn't give equal coverage to, say, modal logic and first-order logic in this article

2758:

problematic. It's not problematic at all — it asserts that there is no natural number having a certain primitive-recursive property, and in fact, there is no such number, because if there were, the theory would be inconsistent, contrary to hypothesis.

1879:

Pcap proposed this merger 6 months ago. If I understood everything correctly, then everybody agreed, except JoesphPGrant, who created an account to remind us that some older people are more likely to look for the present article under "symbolic logic".

2887:, then that's abundantly enough Platonism to guarantee that the Goedel sentence of a consistent theory is true. If you're not a realist about models, then I think you owe some sort of account of what you mean when you drag them into the discussion. -- 474:

Although this article is Top-priority, it's really barely more than a stub. I'm going to give this article the thorough reworking it needs to get to the quality it should be. Please feel free to help... and don't be surprised at the changes. — Carl

5307:

I think a complete rewrite of the sentence is indicated. We should try to be clear that the boundaries are fuzzy, and avoid trying to capture the essence of mathematical logic by some phrase that sounds more precise than the concept actually is.

4996:

article. As far as I know, Russell's paradox is a set theory issue, not a first-order language (predicate calculus) logic issue. The paradox is resolved in ZF by the regularity axiom, and it is resolved in NBG by distinguishing sets from classes.

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Either you specify a formal theory able to be meta-arithmetic (it could be ZF-like but with only finite sets), or you use arithmetic itself, via the Gödel encoding (of statements and proofs by natural numbers). It looks like a computer code of 40

2979:

Second, mathematicians don't use "true" to mean "provable in the standard model". They use it (for arithmetic statements, anyway) to mean "true in the standard model", which is different. (In fact, it's precisely the Gödel results that show it

1080:

in medieval and then 19th century work is, most likely, simply because non-Western work was much less known at the time. The same pattern has repeated itself in many areas of mathematics, where there was much duplication of effort (for example,

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No, I disagree. I think these are central issues to the topic and need to be addressed in the main text. There is not some unspoken universal understanding that everyone has of all this terminology. And I can't speak for all formalists, but

5241:"a figure of speech through which the act of cognition is designated, reflecting the objective difference between real objects and elements of consciousness; In formal logic, one of the logical devices that can be used instead of defining". 1544:

is, but the claim that it's purely about syntactic relationships, as the lead currently says, I think is just false. My understanding is that it's a mostly-disused phrase for mathematical logic, surviving in traditional titles such as

2930:

Also — it is a standard convention in mathematical discourse to talk like a realist whether you are one or not. That convention should be observed here too. An explanatory note could be used to give an overview of the deeper issues.

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Well, I hoped that somebody had given a reasonable mathematical definition for what I would call a "logic", but I am beginning to suspect that I was wrong. "Logical systems" in Lindström's theorem come close to what I mean, and so do

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Exactly what these fields have in common is a little hard to put your finger on, but I'm not sure that "applying formal logic to mathematics" is really it. In lots of cases it's actually the reverse, applying mathematics to formal

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As for me, it is a pity that the "nice example above" is scantily explained in Knowledge. A special article about "True but unprovable" would help against misunderstandings with models, axioms, truth etc. (Likewise, I volunteered

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I copied my working draft here, so that other people can contribute. It is not by any means complete; many paragraphs are just sketches. I plan to add references for all the years in parentheses, just haven't typed them in yet.

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subfield which roughly corresponds to our "formal logic" section (which has a silly heading because all mathlogic is formal). In MSC2010 this is considered to contain quite a bit more stuff than what's mentioned here, including

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in set theory and mathematical logic, and that none of these theories can be formalized without the other. Trovatore's POV seems to reflect a fairly high base line ( or high set bar for those of you into high-jumping) , while

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I'd like you to stop doubting so much and instead accept as empirical fact that not everybody will be on the same page there. I'm not even asking you to abandon language like "true but unprovable," just to say what it means.

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Carl: thanks for your excellent precis. It may be best to duck stating the relationship, since there is controversy over whether categorical logic is algebraic logic or some quite different way of doing logic algebraically.

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runs over the well-known formalizations of arithmetics). I definitely prefer models that embed into models of ZFC; even better, of ZFC with some large cardinals. In this sense I am much more platonist than a 100% formalist.

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to symbolic logic in this article... You have deleted the entire subject on Knowledge... As restricted intellectually, as the topic is, it should still be a part (article entry) of Knowledge... Hey, maybe we can merge

1346:

885:

I am feeling a bit guilty that I haven't revised the section on model theory otherwise, as I meant (and promised) to do. I find it very hard to describe what I think is a large, heterogeneous subject in just a few words.

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model. As a formalist, it's rather bizarre to hear it claimed that sentences in a formal language are "about" anything, in particular - they're just strings. Nothing in modern math requires that they be descriptive.

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The problem with this line of discourse is that it leads into the narrative, "oh, well, maybe it's a nonstandard model that's correct, who's to say?". But that's not really a coherent view. If you're a realist about

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Interestingly, Gödel also investigated closed time-like curves in models of space-time. I guess he was thinking about (at least) two ways to know the future: predict it by math, or observe it, if it is also the past.

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Is modal logic really established as part of mathematical logic? Given the ease with which Kripkean modal logics can be expressed in first-order logic, the case for modal logic is not directly one of expressiveness.

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Well they are all valid forms of mathematical logic, but the article overall has far too few references. It really is a bit of a mess that way. I think I'll add some tags on sections which have no citations at all.

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For almost every intent and purpose, "mathematical logic" simply means the union of proof theory, recursion theory, model theory, and set theory. None of the latter three of those could possibly be called "symbolic

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some default foundational assumptions than agreeing that formal sentences have some natural-language interpretations that justify privileging some models over other models (the designation of a "standard model" is

1507: 963:. You've been bold, and been reverted. Now it's time for you to see if you can gain consensus. Your orders not to revert, in your edit summaries, are not going to accomplish anything, except piss people off. -- 728:

I have done all the obvious or trivial changes immediately after proof-reading. Now I will soon start with a few things where I am not entirely sure what to do. You might want to have a look at them afterwards.

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I don't believe that, today, category theory is considered part of mathematical logic by either category theorists or mathematical logicians. Perhaps in 50 years, it will be, but that's hard to predict. — Carl

2859:

Okay, correct me if I'm wrong because I'm just a stupid geometer, but I think the situation here is that we have a statement about a theory that can interpret "arithmetic," and when the model of arithmetic is

2534:. I generally think of "symbolic logic" as being topics like Venn diagrams, not what we usually think of as mathematical logic. The Mathworld article "Symbolic Logic" cited as the the only reference for the 2082:, by the way. There's opposition to this effort, I can't find a link off the top of my head, but some mathematicians wrote that mechanized proofs are often uninsightful, so such projects are a waste of time.) 4115:

Naive reader: What the hell is happening here? I needs facts about groups. Can I know them without learning all these logics? Could these persons, with their exotic orientations, form a separate club, please?

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As a researcher in mathematical logic, if someone told me they studied "logic", I would begin by asking what department they work in, because studying "logic" on its own does not mean very much to me. — Carl

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This is similar to the argument raised by Rick above (to argue that symbolic logic is not necessarily the same as mathematical logic.) It's true that natural language proofs actually dominate in mathematics;

2479:... Hey, what do you know? If we keep doing this, we can really condense Knowledge down to a more "manageable" level... Wow, someone really needs to return to college... The entire article that was here on 1338:... Hey, what do you know? If we keep doing this, we can really condense Knowledge down to a more "manageable" level... Wow, someone really needs to return to college... The entire article that was here on 3825:

was confused by the way these statements were written - I knew there must be some underlying platonist assumptions going on, but it wasn't clear to me what they were because I don't think that way. --

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like proof theory or model theory, which philosophers would call metatheory or metalogic. I don't think Britannica is a good example to follow for organization here, but they take that approach; see

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Sorry, no, it has little to do with what most mathematicians "are". It has to do with Gödel's theorem and maybe about what Gödel "were" (Platonist, Realist, Formalist, ...). (He wasn't a goof-bag.)

2053:), and more obscure topics in the rather comprehensive MSC shouldn't even be mentioned here. I was merely asking whether some of the current structure/contents is by design or by accident/omission. 3219:

Well I believe that you don't believe me, and would never accuse you of deceiving yourself out of unwillingness to come to terms with the fact that some people don't believe in abstract objects. --

3659:

theorem. "Generalized" statements, using different premises, about it should not refer to it as Gödel's theorem, but rather a generalized version. For this, you absolutely certainly need sources.

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logic of avoiding philosophical rambling. But I have some leads for history books that might prove useful. I expect that once I copy the new version here, other people will round out the coverage.

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Hans, thanks for your help this afternoon. The article is, as everyone can see, still very bare-bones with very little exposition. I am adding references, and will eventually convert them to the

5370:(First link in the article), It comes up with disgusting unrelated image attached to the article. Please fix it! I don't know how to sorry. It appears it was griefed or the image was misplaced. 1413: 2988:

you need to find out that the "standard model" is actually just the one you had in mind all along (you know, zero, one, two, three, etc), but explained with all this superstructure around it.

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number - that becomes a very unintuitive counterfactual and I'm not sure it's even meaningful. But let's say you accept this. Obviously you can't test it directly, so you then say, okay, it

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Many-valued logics should be added to the section "Nonclassical and modal logic", and type theory should be added to the "formal logics" section. It's hard to remember everything at first.

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most of the material is a kind of essay which mostly refers to classic papers, not so much to references which support the assertions which are made about history and the current subject.

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I don't see how model theory and (even worse) recursion theory can be said to study the metatheory of formal proofs? Mathematical logic includes more than proof theory; see below. — Carl

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First, that isn't how we know it's neither provable or disprovable within the theory; having proved that it's neither provable nor disprovable in the theory, we then conclude (using

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Formalist A: The article assumes implicitly that groups exist. But no; what really exists, is proofs of theorems within the group theory. Thus, we must specify "group theory". Like

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I wouldn't separate constructivism out as a separate area, really. Constructivists have their own versions, even if they're sometimes scarcely recognizable, of all four areas. --

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The terms of equational logic are built up from variables and constants using function symbols (or operations). Identities (equalities) of the form s=t, (1) where s and t are ...

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The proof theories of propositional calculus and first-order logic are often referred to as classical logic. Intuitionistic propositional logic can be described as classical ...

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There was a time when Knowledge explicitly wanted to have an article on every subject that Wolfram Mathworld had an article on. Here, fyi, is a list of their articles on logic:

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I've gone ahead and rewritten it, using text from the other article. But I'm not at all attached to the new version, so please rewrite it again if you see room for improvement!

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which would seem to prove that 4=5 ;-). Can someone straighten this out? There are two obvious possible solutions and I defer to the experts to figure out which is better.

1630:

In mathematics, a formal language is normally defined by an alphabet and formation rules. The alphabet of a formal language is a set of symbols on which this language is ...

4745:"we don't need to generate absolute agreement on the nature of mathematical truth here, only arrive at an acceptable wording for the article." Give this person a medal. -- 1785:

Symbolic/formal logic: "The study of the meaning and relationships of statements used to represent precise mathematical ideas. Symbolic logic is also called formal logic."

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In this framework it becomes true that Goedel's statement is unprovable (in T) but true (provable in some T1 that is "better" than T in the sense vaguely expressed above).

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The branch of formal logic, also called functional calculus, that deals with representing the logical connections between statements as well as the statements themselves.

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The set of terms of first-order logic (also known as first-order predicate calculus) is defined by the following rules: 1. A variable is a term. 2. If f is an n-place ...

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The formal mathematical study of the methods, structure, and validity of mathematical deduction and proof. In Hilbert's day, formal logic sought to devise a complete, ...

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would be mathematical logic as it was practised in roughly the first half of the 20th century. If we ever need to split this article per summary style, or need to split

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Restricting from "arithmetic" to "the natural numbers" (a specific model) is what I meant by going to a larger set of rules. And obviously this has caused confusion. --

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A syllogism, also known as a rule of inference, is a formal logical scheme used to draw a conclusion from a set of premises. An example of a syllogism is Modus Ponens.

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A fundamental system of logic based on the concept of a generalized function whose argument is also a function (Schönfinkel 1924). This mathematical discipline was ...

938:. Note that even if you find a source or two, it's not good enough if it's a fringe view (though it would be reasonable in that case to mention it as a minority view). 5403: 4596:

No, that's trivial — anything can be proved by allowing more axioms, if the axioms are not in any way constrained. The important point to get across here is that, if

2511:

Whom you are addressing, and in what way you connect that person's actions with party affiliation, is obscure to me. As I've said above, my understanding of the term

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which is the logic of mathematics... Are you a Republican? Way to confuse the topics... Maybe if you do a search on ancient civilizations, you will only come up with

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which is the logic of mathematics... Are you a Republican? Way to confuse the topics... Maybe if you do a search on ancient civilizations, you will only come up with

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Unless that topic is a different field of study somehow (can't tell from the stubby article), I propose it be turned into a redirect here and mentioned as a synonym.

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on Knowledge. If you would like to support the project, please visit the project page, where you can get more details on how you can help, and where you can join the

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accurate about the content. For example, if a book from 1940 was republished in 1990, it's still not going to have information on results proved after 1940. — Carl

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theorem, but some other (slightly more general) theorem. I'm quite sure Gödel did not have any goofy, yet consistent, theory in mind when he did his original work.

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is it ever a pre req for other math classes like finite math. Would geting a good grade in mathematical logic convince a counselor to let you skip certain classes?

1291:- oh god, not again! We need some final resolution of this wikisore, I guess an RfC. I don't have either the appetite or time to put one together soon, though. — 1389: 1386: 5418: 3733:

There is a difference between statements in ordinary language and in mathematical logic. (English is decidedly not consistent. interesting, but not helpful here)

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And why the Fk are the posts ITT intersected so that my early reply appears to be the last one? Just delete it if you aren't interested in other peoples opinion,

787:

Cantor first appears in this article in the discussion of the well-ordering principle. Shouldn't his contribution be mentioned earlier in the history section?

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obvious that, at least in this case, the axiomatization has a specific intended interpretation, namely the naturals with ordinary addition and multiplication.

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Contemporary mathematical logic is roughly divided into four areas: set theory, model theory, recursion theory, and proof theory and constructive mathematics.

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And yes, matheworld's definition of "mathematical logic" is wrong there, unless recursion theory has suddenly morphed into the study of formal proofs. — Carl

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Example: What does you mean when you say that every natural number is the sum of four squares? Do you mean that there's a formal derivation of the string

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I agree that that's a problem. For me the balance was only slightly in favor of 2001, so I am not surprised you prefer 1967. I have changed the footnote to

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must in fact hold. If they did not, there would be a witness, and that witness would code a proof that could be turned into a proof of a contradiction in

2769:

of arithmetic statements is the same as truth in the standard model. The fact that the nonstandard models are "useful" has nothing to do with anything. --

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We use this language in mathematics articles all over the place. There's nothing essentially diffrerent here. We should follow the same convention. --

2864:, it is true, but when the model of arithmetic is the non-negative hyperreal integers, it is false. Is that right? If so, could the article say that? -- 2845:

I'm not sure I follow you. What do you mean "it satisfies this theory but not that one"? I can't tell what "it", "this theory", or "that one" are. --

3938:

for an example of an article that gives a brief treatment of its topic without omitting the definitions or relying on unclear, ambiguous intuitions. --

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function to any number off the critical line and not a negative even integer, the result is nonzero. (Try it!) Unless what you really mean is that you

2995:

mean ordinary addition and multiplication, and it all boils down to an arithmetic statement is "true in the standard model" just in case it's ... true.

2287:

minimalist tradition, that's not the same thing as saying they're constructivists; conversely, constructivists are by no means limited to proof theory.

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superior to Mathworld by now, and most of the time when Mathworld shows up, it's to cause trouble (usually, someone copying some silly MW neologism).

4230:

I see. And in the case of the parallel line postulate I agree that it should not be treated as true for the physical space. But on the other hand,

1638:

The study of the meaning and relationships of statements used to represent precise mathematical ideas. Symbolic logic is also called formal logic.

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The table is from (Frank Pfenning - Proof Theory Foundations, Lecture 1, Oregon Programming Languages Summer School 2012, University of Oregon)

5239:, but I would appreciate someone with more expertise reviewing this. My initial description is based entirely on the Russian Knowledge article: 990:

This doesn't mean that education, law, manufacturing, archaeology, health, psychology, music, military etc. are branches of computer science. --

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template. Many of the sections could use rearranging if not complete rewriting. And the history from 1935 to 1950 is almost nonexistent. — Carl

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development here in the past few months or so, there's no consensus yet in the model theory community about the status of Vaught's Conjecture.

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Yes, certainly, a synopsis of Cantor's work in set theory needs to be added to the 19th century section before the article is complete. Also

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sources). I'm not used to writing "nontechnical" articles like this, so it's an experiment for me to find an acceptable presentation. — Carl

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If this table really does belong here, it's going to need some sort of explanation. A search for "Lax logic" in wikipedia doesn't find much.

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Oh, really? Well, but maybe its place is in another article - about Proof theory, from the viewpoint of computer science rather than math?

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arithmetic could be of some interest. But a nonstandard model of meta-arithmetic is an exotic idea. Saying "if the machine stops at a time

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This discussion above begins: "The Goedel sentence of a consistent theory is true. That is what reliable sources say," and assuming that

2139:

Actually, most of the time I think "mathematical logic" means "symbolic" or "formal" logic, and when it doesn't people just say "logic".

3392:(Nice example above.) Even if the suggested (and multiply reverted) formulation was to be taken in, then we would not be talking about 2336: 2270: 984: 1716:

in the first place, but it certainly is not a definition of "mathematical logic". So I agree with Trovatore's assessment above. — Carl

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same thing? very different? do we need 'see alsos' between the two? (I don't know for sure since I only studied it as an undergrad.)

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At the moment category theory is mentioned but not in the guise of categorical logic (to be fair, we also don't mention linear logic,

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category theory and perhaps universal algebra, but these are less commonly listed). None of these fields are "called formal logic".

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But I think you are missing the point that Goedel's original work is not the only reliable source on the incompleteness theorems. --

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A premise is a statement that is assumed to be true. Formal logic uses a set of premises and syllogisms to arrive at a conclusion.

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In this case, saying "the Gödel sentence of a consistent theory is true" is just the natural-language way of expressing the claim

1974:

I see that algebraic loigic is actually mentioned, but in the "formal logic" section. Speaking of which: the classification has a

848:

that the counterexample is correct. I think the proposed example should not be mentioned in this article, and I will remove it. --

2973: 1788:(mathematical) logic: "The formal mathematical study of the methods, structure, and validity of mathematical deduction and proof. 219: 174: 5204:, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section. 4582:

with a certain smaller set of rules." We can say that instead of just letting people wonder what "true but unprovable" means. --

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of arithmetic and set theory that we are talking about. We can do all that without debating whether some objects are "real." --

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This is akin to claiming "the parallel line postulate is 'true'" with the justification that "the parallel line postulate is

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is a situation when he faces a statement with a supposedly default meaning for the first time as a professional formalist.

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Better overview of logic besides first order logic, more thorough history, the technical reference should be split out.

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logic. I feel that considerations of "user friendliness", particularly towards older readers, weigh against a merger.

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in its old place. It doesn't seem to be very relevant to anything other than the history of the merged article itself.

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While I have no strong opinion one way or the other about the merge, there is a difference between formal logic, as in

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using these tags a few days ago, this would have resulted in something like (2001), but that's no longer the case. --

4992:

This doesn't sound right to me at all. I've never read this anywhere, and the name "Russell" does not appear in the

4764:

Can someone explain how this new table in the "Formal logical systems" section is connected to mathematical logic?

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It is true that there are nonstandard models of arithmetic satisfying the negation of the sentence. But these are

1807:

Let's please not be relying on Mathworld for, well, anything really. The corpus of mathematical articles on WP is

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books), it appears that symbolic logic is the former/traditional name given by philosophers to mathematical logic.

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It would help if you could identify, on this talk page, some claims that seem false or tendentious to you. — Carl

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needs to be mentioned. Thanks for reading through the article and giving other suggestions, or editing it. — Carl

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study models of arithmetic (an interesting topic in its own right, to be sure, but a different one); it studies

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My opinion, and that of many published reliable sources, is that there is no God and there are no integers. Per

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is not controversial; formalists can be comforted that it's formally provable in even a very weak theory (say

3136:, obtainable by starting with axioms of Peano arithmetic and Hilbert logical axioms and applying modus ponens? 2471:

and think that all earlier civilization were founded by extraterrestrials... Then we can merge the 2 articles

1330:

and think that all earlier civilization were founded by extraterrestrials... Then we can merge the 2 articles

1200:.). Should we have a paragraph on categorical logic? I'm not sure yet, so the answer is probably yes. — Carl 5176: 5139: 5099: 5062: 5007: 4873: 4840: 4750: 4705: 4201: 4078:

Formalist D: Do not bother; these two are well-known to be equivalent. You'd better think, whether you mean

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instead of posting on article talk pages; (b) explain your proof strategy in English before formalising it.

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a branch of computer science? It's hardly compelling to use one WP article as a source for another. — Carl

601:. But I think it spends too much time on symbolic logic, which is only part of mathematical logic. — Carl 5081: 3708: 2653: 2568: 2551: 2516:

understanding, one that you can make precise and for which you can give references, by all means share. --

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Modal logic is part of logic. Indeed, a significant part of Aristotle's work on logic is devoted to it.

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because it will otherwise sound like gibberish to people like me and the IP user who commented above. --

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area besides those four you've mentioned. It's true than many of those are studied with respect to some

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part of "mathematical logic"; it may be that there is simply no better place to put that topic. — Carl

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There is no page on Formal Logic! It redirects to this page, and I believe it is missing a huge part.

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Conclusion: Either someone's got to back up the assertion with a reference, or it has to be corrected.

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is true" just means "for every natural number blah blah blah", and the "true" part is redundant (see

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This presupposes that the idea of a "correct" model is a coherent one. This is a contentious claim.

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in that way, the title is available for that purpose. But I don't think that's going to happen soon.

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The Goedel sentence is an arithmetic statement. That means that the natural numbers are what it is

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on Knowledge. If you would like to participate, please visit the project page, where you can join

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It doesn't have a citation and is far too much without a citation so it should just be removed.

3964:

OK, I take your challenge about "groups". Let us imagine such discussion on the talk page there:

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to talk about plane geometry." I find it intensely confusing, and I doubt I am the only one.

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Can you point out any source, apart from that wikipedia list, that claims mathematical logic

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logic. So "formal logics" are the ones in which we have a notion of a formal proof. — Carl

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This 2008 philosophical encyclopedia says that mathematical logic includes symbolic logic.

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First-order logic is a kind of formal system of logic designed to avoid Russell's Paradox.

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Talk:Mathematical logic#Was first-order logic really designed to avoid Russell's Paradox?

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Being a formalist, face the consequences yourself, and do not make troubles to innocents.

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I'm not sure exactly what you're saying, but I'm of the view that this article is indeed

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This deserves a section. It may be best to treat it together with categorical logic. —

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The indentations indicate who is responding to whom. I wouldn't take it that no one is

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On the face of it, though, the claim is just obviously wrong. Math logic considerably

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Thanks. I edited it for clarity but feel free to revert if you think I went too far.

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The version on French wikipedia isn't bad. You can get google to translate it for you

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Whether Knowledge still wants to have an article on all these topics, I do not know.

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I agree. Thank you for getting in touch with me, and thank you for the correction.

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think your way or mine, it's best to explain the details of what you really mean. --

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I don't think we need more see also links, but I added a link from the text here to

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Note that your edits will be reverted until you can provide any reliable sources!

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I'm delighted to see the improvement in the article, and have just one suggestion.

448:. For the contribution history and old versions of the redirected page, please see 5220:

Last edited at 20:43, 16 April 2007 (UTC). Substituted at 20:05, 1 May 2016 (UTC)

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will never find the proof of the TRUE statement that it will never stop! See also

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How is epistemic/temporal/linear/lax/modal logic connected to mathematical logic?

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against misunderstandings with affine spaces etc.) I tried once something alike:

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This article (written in 2000) is pretty interesting and may be worth a mention:

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But I think we can take Hilbert's and Ackermann's word that it's the same topic.

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is consistent, then the relationships among the natural numbers expressed by

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Perhaps you are too exclusionary? After all, MSC does include a fairly beefy

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Lack of current references for the history and current state of the subject.

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space; taken this way, it gets a status closer to that of natural numbers.

2496:- they sort of look the same... then delete Calculus and we're all set... 2261:

Is mathematical logic consistent? The "Subfields and scope" section says:

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I got bold and deleted the table. If someone wants to defend it, let him.

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satisfying the hypotheses. Just taking the troublesome part by itself, "G

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out into what it says, "for every natural number blah blah blah", then "G

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I made an initial effort to provide a description and properties of the

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in your comment. Just no one has responded to it (as yet, anyway). --

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slightly broader than the average mathematical logic textbook". — Carl

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calculus would be a branch of physics, economics, biology, etc. — Carl

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so I apologise in advance if there is some insight to be found there.

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You are right. Both about indentations and about the disputed matter.

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words to say explicitly it satisfies this theory but not that one? --

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Samuel R. Buss, Alexander A. Kechris, Anand Pillay, Richard A. Shore.

1454:. But he makes no mention of any other branches of symbolic logic... 195: 168: 5326:, both as something to borrow from and perhaps to improve as well! 4556:

Oh, and in case it's not obvious, I forgot to add: The conclusion

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formalist, while humans are not. (By the way, did you ever try a "

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Was first-order logic really designed to avoid Russell's Paradox?

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Wow! Then, I have nothing to say you. We are "mutually bizarre".

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Halting problem#Relationship with Gödel's incompleteness theorems

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When I hover over the Mathematical Logic link to this page from

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http://math.ucsd.edu/~sbuss/ResearchWeb/FutureOfLogic/index.html

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assumption isn't that easy to put in practice; see for instance

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things that could be added to the model theory sections? — Carl

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I don't see why Goedel's view should be given special weight...

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http://math.ucsd.edu/~sbuss/ResearchWeb/FutureOfLogic/paper.pdf

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is entitled "Proof theory and constructive mathmatics". — Carl

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This merge seems OK to me. Honestly I don't know exactly what

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and Russell; omitting that omits the key parts of its history.

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defined mathematical logic as only a branch of symbolic logic

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Wapedia is a Knowledge mirror. You probably meant to refer to

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Well, for a formalist, every theorem is a theorem of a given

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I reverted the entire section to a previous version. — Carl

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Mathematical logic is not a branch of computer science. Our

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My goal is to end up with an article that can be put up for

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Excellent! You removed some other dubious material as well.

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hey-neither-really-true-nor-really-false-wow-man-heavy. --

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I am the ghost of Ludwig Wittgenstein. See me after class.

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ways to communicate our ideas. Also formal languages are

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Well, that's exactly what "true" means in this context. --

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See for example the previous section of this talk page:

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theory, and could make a case for universal algebra). --

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what reliable sources say, that's where it should end.

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His theorem should give him 100% weight because it is

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but not much as a description for current research. --

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computer science, so it can't be a "branch" of it. --

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https://www.youtube.com/watch?v=YRu7Xi-mNK8?t=13m24s

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curves go beyond my imagination. Our article on the

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mathematical logic a bit differently. In particular

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http://wapedia.mobi/en/Outline_of_computer_science#1

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Well, some of your details are not quite right :-).

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Very good. I just noticed that the French article (

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Nice job with merging 1951:is considered a separate subfield containing 1415:this philosophy book gives them as synonyms. 1322:, which is the logic of language meaning and 1318:Absolutely, amazing... Nice job with merging 409:link inline references to publication details 8: 2687:If mathematical logic is a branch of math..? 2530:I don't see any "symbolic logic" article at 2099:echoes what I wrote in the above paragraph. 4820:Generic effects (monads, state, exception) 2706:You might consider asking this question at 2423:Journal of Symbolic Logic 7 (2001) 169-196. 454:; for the discussion at that location, see 4798:Reactive Programming (partial evaluation) 4766: 1734:Not sure how they were in 2007, but today 574: 387:Here are some tasks awaiting attention: 272: 163: 58: 3037: 2950:, and is false for some other model) and 4053:? That is your POV. As for me, I prefer 3985:First-order logic#Ordered abelian groups 3470:may be a kind of hyper-integer. Do you? 1642:Intuitionistic Logic (Wolfram MathWorld) 5404:Knowledge vital articles in Mathematics 5224:Help with term: distinction on Wikidata 4958: 165: 60: 19: 5240: 4980:This currently appears in the section 2739:2A01:E35:8AD5:C150:790F:3BD7:736A:B759 2595:propositional calculus = formal logic? 2452:Symbolic Logic and Mathematical Logic 1674:Predicate Calculus (Wolfram MathWorld) 233:about philosophy content on Knowledge. 5419:B-Class vital articles in Mathematics 5368:https://en.wikipedia.org/Peano_axioms 1666:First-Order Logic (Wolfram MathWorld) 1658:Combinatory Logic (Wolfram MathWorld) 908:it's a branch of CS as well. source: 7: 3166:number if you actually tried it for 1793:A distinction without a difference? 1650:Equational Logic (Wolfram MathWorld) 1385:Based on a few books I've looked at 1243:Sorry about that. The right link is 217:This article is within the scope of 106:This article is within the scope of 4685:, which could be something else. -- 2188:my post on Arthur Rubin's talk page 1891:In my opinion the natural scope of 1626:Formal Language (Wolfram MathWorld) 1506:, to agree that they are the same. 1485:Church also says they're the same. 985:ACM Computing Classification System 398:information in references at bottom 49:It is of interest to the following 5439:Mid-importance Philosophy articles 4523:is true" says nothing other than G 3063: 3057: 3051: 3045: 3039: 2916:bring them into the discussion. -- 1634:Symbolic Logic (Wolfram MathWorld) 1219:Mathematical logic#Algebraic logic 14: 5429:Top-priority mathematics articles 5202:several discussions in past years 4527:itself. That is, if you expand G 2726:Definitions of mathematical logic 1814:In this case, MW's definition of 1355:- they sort of look the same... 126:Knowledge:WikiProject Mathematics 5459:Knowledge pages with to-do lists 5399:Knowledge level-4 vital articles 5197:Talk:Mathematical logic/Comments 427: 378: 328: 239:Knowledge:WikiProject Philosophy 204: 194: 167: 129:Template:WikiProject Mathematics 93: 83: 62: 29: 20: 1782:Mathworld defs for comparison: 1502:We can also get a philosopher, 1126:modality is a valuable tool? — 653:Bolzano, Cauchy, Epsilon, Delta 259:This article has been rated as 242:Template:WikiProject Philosophy 146:This article has been rated as 5409:B-Class level-4 vital articles 5173:lists of dubious passages too! 4384:ways to communicate our ideas. 3123: 3069: 2369:, you should (a) take this to 2345:02:36, 24 September 2009 (UTC) 2327:10:19, 21 September 2009 (UTC) 2312:Handbook of Mathematical Logic 2300:07:02, 21 September 2009 (UTC) 2279:05:33, 21 September 2009 (UTC) 2247:10:41, 21 September 2009 (UTC) 2214:Handbook of mathematical logic 2200:03:44, 21 September 2009 (UTC) 2174:19:25, 20 September 2009 (UTC) 2149:14:52, 20 September 2009 (UTC) 2126:19:26, 20 September 2009 (UTC) 2109:16:41, 20 September 2009 (UTC) 2092:14:49, 20 September 2009 (UTC) 2063:14:29, 20 September 2009 (UTC) 2043:13:38, 20 September 2009 (UTC) 2002:11:35, 20 September 2009 (UTC) 1969:11:10, 20 September 2009 (UTC) 1868:10:48, 21 September 2009 (UTC) 1851:10:43, 21 September 2009 (UTC) 1833:05:19, 21 September 2009 (UTC) 1803:05:03, 21 September 2009 (UTC) 1778:04:57, 21 September 2009 (UTC) 1756:14:16, 20 September 2009 (UTC) 1729:19:36, 20 September 2009 (UTC) 1707:14:34, 20 September 2009 (UTC) 1595:14:11, 20 September 2009 (UTC) 1573:23:19, 19 September 2009 (UTC) 1559:23:16, 19 September 2009 (UTC) 1536:11:54, 19 September 2009 (UTC) 1519:11:48, 19 September 2009 (UTC) 1498:11:40, 19 September 2009 (UTC) 1481:11:30, 19 September 2009 (UTC) 1464:11:26, 19 September 2009 (UTC) 1443:09:01, 19 September 2009 (UTC) 1425:08:54, 19 September 2009 (UTC) 1408:08:48, 19 September 2009 (UTC) 1380:08:39, 19 September 2009 (UTC) 959:On another note, the cycle is 1: 5449:Mid-importance logic articles 5362:Hover over Mathematical Logic 5273:20:31, 13 November 2020 (UTC) 5252:20:28, 13 November 2020 (UTC) 5025:. Maybe, try to contact him. 3774:too vague and should specify 3418:"A theory as a crystal ball?" 2747:11:49, 12 November 2013 (UTC) 2639:12:50, 14 February 2011 (UTC) 2617:04:35, 14 February 2011 (UTC) 1690:Syllogism (Wolfram MathWorld) 660:23:32, 30 November 2007 (UTC) 642:21:10, 30 November 2007 (UTC) 614:17:16, 30 November 2007 (UTC) 589:04:24, 23 December 2014 (UTC) 564:16:20, 30 November 2007 (UTC) 543:14:51, 29 November 2007 (UTC) 528:scientific citation guideline 526:, which includes meeting the 504:16:55, 26 November 2007 (UTC) 488:15:59, 26 November 2007 (UTC) 120:and see a list of open tasks. 5424:B-Class mathematics articles 2974:Gödel's completeness theorem 2447:06:50, 7 November 2009 (UTC) 896:11:00, 11 January 2008 (UTC) 877:00:15, 11 January 2008 (UTC) 820:14:38, 3 December 2007 (UTC) 794:14:25, 3 December 2007 (UTC) 766:11:37, 3 December 2007 (UTC) 756:03:57, 3 December 2007 (UTC) 734:18:31, 1 December 2007 (UTC) 716:16:35, 1 December 2007 (UTC) 698:09:56, 1 December 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Mathematics 5444:B-Class logic articles 5394:B-Class vital articles 4787:Distributed Computing 4376:Natural languages are 3264:User talk:128.8.140.59 3130: 2485:not a single reference 2454:are not the same thing 2360:and deleted by poster) 1344:not a single reference 297: 220:WikiProject Philosophy 4086:logic. And, why just 3466:" I do not mean that 3131: 2409:logic in 21st century 434:The contents of the 296: 36:level-4 vital article 3920:" software? I did.) 3634:continuum hypothesis 3036: 2473:Ancient civilization 2469:Chariots of the Gods 1981:substructural logics 1332:Ancient civilization 1328:Chariots of the Gods 132:mathematics articles 4792:A is true at time t 4565:Robinson arithmetic 4285:group (mathematics) 3936:group (mathematics) 3353:True but unprovable 2624:propositional logic 2544:philosophical logic 1924:Talk:Symbolic logic 1885:Talk:Symbolic logic 1287:Propositional logic 1253:Propositional logic 961:bold-revert-discuss 826:Vaught's Conjecture 465:Archived discussion 245:Philosophy articles 5324:the text over here 5190:Assessment comment 5177:Alan U. Kennington 5140:Alan U. Kennington 5100:Alan U. Kennington 5063:Alan U. Kennington 5008:Alan U. Kennington 4874:Alan U. Kennington 4841:Alan U. Kennington 4092:Higher-order logic 4026:rules of inference 4012:and list not just 3126: 2952:that's how we know 2465:mathematical logic 2367:Talk:Number theory 2351:Collatz conjecture 1908:Mathematical logic 1902:I have redirected 1816:mathematical logic 1766:mathematical logic 1760:Also, they define 1324:mathematical logic 1277:The right link is 446:Mathematical logic 368:Updated 2007-12-11 339:Mathematical logic 298: 230:general discussion 101:Mathematics portal 45:content assessment 5261: 5237: 5218: 5217: 5165: 5124: 5051: 5023:User:Rick Norwood 4994:First-order logic 4982:First-order logic 4835: 4834: 4770:Logical Judgement 4650: 4094:can be used, too. 4088:first-order logic 4051:natural deduction 3376:Quine (computing) 2984:to be different.) 2737:comment added by 2637: 2615: 2546:instead of here. 2477:Extraterrestrials 2361: 2325: 2245: 2197: 2172: 2124: 2106: 2089: 2060: 2041: 1999: 1985:many-valued logic 1966: 1953:categorical logic 1866: 1849: 1800: 1775: 1753: 1727: 1711:"Symbolic logic" 1565:Pontiff Greg Bard 1533: 1516: 1495: 1478: 1461: 1440: 1422: 1405: 1377: 1336:Extraterrestrials 1299: 1285:is a POV fork of 1266: 1251:is a POV fork of 1211: 1189: 1153: 1134: 1095: 1082:Pascal's triangle 1067: 1036:comment added by 1022: 918:comment added by 875: 818: 754: 714: 679: 640: 612: 591: 579:comment added by 541: 486: 462: 461: 422: 421: 416: 415: 319: 318: 315: 314: 311: 310: 307: 306: 212:Philosophy portal 162: 161: 158: 157: 5466: 5351: 5346: 5335: 5330: 5279:Opening sentence 5259: 5235: 5207: 5206: 5199: 5155: 5114: 5041: 4968: 4963: 4767: 4643: 4055:sequent calculus 3135: 3133: 3132: 3127: 2753:Goedel sentences 2749: 2627: 2612: 2605: 2600: 2538:(now merged) of 2355: 2315: 2235: 2195: 2162: 2114: 2104: 2087: 2058: 2031: 1997: 1964: 1897:history of logic 1856: 1839: 1798: 1773: 1751: 1717: 1531: 1514: 1493: 1476: 1459: 1438: 1420: 1403: 1375: 1295: 1256: 1201: 1185: 1143: 1130: 1085: 1057: 1048: 1012: 930: 865: 808: 744: 704: 669: 630: 628: 622: 602: 531: 476: 453: 431: 430: 424: 382: 375: 374: 369: 332: 331: 321: 283: 273: 247: 246: 243: 240: 237: 214: 209: 208: 207: 198: 191: 190: 185: 182: 171: 164: 134: 133: 130: 127: 124: 103: 98: 97: 87: 80: 79: 74: 66: 59: 42: 33: 32: 25: 24: 16: 5474: 5473: 5469: 5468: 5467: 5465: 5464: 5463: 5384: 5383: 5364: 5349: 5344: 5333: 5328: 5281: 5226: 5195: 5192: 5093: 5027:Boris Tsirelson 4978: 4973: 4972: 4971: 4964: 4960: 4940:Boris Tsirelson 4908:Boris Tsirelson 4803:A is a resource 4784:Epistemic Logic 4773:Branch of Logic 4762: 4605: 4561: 4534: 4530: 4526: 4522: 4513: 4461:Boris Tsirelson 4336:Boris Tsirelson 4237:Boris Tsirelson 4155:Boris Tsirelson 4022:formation rules 3922:Boris Tsirelson 3918:proof assistant 3884:Boris Tsirelson 3757:Boris Tsirelson 3592:Boris Tsirelson 3472:Boris Tsirelson 3422:Boris Tsirelson 3380:Boris Tsirelson 3357:Boris Tsirelson 3034: 3033: 2765:models. Truth 2755: 2732: 2728: 2689: 2668: 2610: 2603: 2597: 2457: 2432:pdf link from: 2411: 2356:(Post moved to 2353: 2259: 1976:"general logic" 1949:algebraic logic 1941: 1315: 1309:Merge/Redirect 1293:Charles Stewart 1279:Boolean lattice 1245:Boolean lattice 1223:Boolean algebra 1183:Charles Stewart 1179: 1176:Algebraic logic 1128:Charles Stewart 1119: 1077: 1031: 936:reliable source 913: 906: 828: 782: 626: 620: 472: 470:Major reworking 449: 428: 418: 417: 412: 343: 329: 281: 244: 241: 238: 235: 234: 210: 205: 203: 183: 177: 131: 128: 125: 122: 121: 99: 92: 72: 43:on Knowledge's 40: 30: 12: 11: 5: 5472: 5470: 5462: 5461: 5456: 5451: 5446: 5441: 5436: 5431: 5426: 5421: 5416: 5411: 5406: 5401: 5396: 5386: 5385: 5363: 5360: 5359: 5358: 5357: 5356: 5340: 5305: 5301: 5280: 5277: 5276: 5275: 5225: 5222: 5216: 5215: 5191: 5188: 5174: 5170: 5169: 5137: 5129: 5128: 5097: 5092: 5089: 5074: 5073: 5059: 5038: 5037: 5005: 4990: 4989: 4977: 4974: 4970: 4969: 4957: 4956: 4952: 4951: 4950: 4921: 4920: 4919: 4918: 4901: 4900: 4871: 4867: 4866: 4838: 4833: 4832: 4829: 4826: 4822: 4821: 4818: 4815: 4811: 4810: 4807: 4804: 4800: 4799: 4796: 4795:Temporal Logic 4793: 4789: 4788: 4785: 4782: 4778: 4777: 4774: 4771: 4761: 4758: 4743: 4742: 4741: 4740: 4739: 4738: 4737: 4736: 4735: 4734: 4733: 4732: 4731: 4730: 4729: 4728: 4727: 4726: 4725: 4724: 4723: 4722: 4721: 4720: 4719: 4718: 4717: 4716: 4715: 4714: 4713: 4712: 4641: 4603: 4559: 4554: 4540: 4532: 4528: 4524: 4520: 4511: 4506: 4484: 4482: 4481: 4480: 4479: 4478: 4477: 4476: 4475: 4474: 4473: 4472: 4471: 4447: 4446: 4445: 4444: 4443: 4442: 4441: 4440: 4439: 4438: 4437: 4436: 4422: 4421: 4420: 4419: 4418: 4417: 4416: 4415: 4414: 4413: 4412: 4411: 4396: 4395: 4394: 4393: 4392: 4391: 4390: 4389: 4388: 4387: 4386: 4385: 4363: 4362: 4361: 4360: 4359: 4358: 4357: 4356: 4355: 4354: 4353: 4352: 4351: 4350: 4349: 4348: 4347: 4346: 4315: 4314: 4313: 4312: 4311: 4310: 4309: 4308: 4307: 4306: 4305: 4304: 4303: 4302: 4301: 4300: 4280: 4258: 4257: 4256: 4255: 4254: 4253: 4252: 4251: 4250: 4249: 4248: 4247: 4217: 4216: 4215: 4214: 4213: 4212: 4211: 4210: 4209: 4208: 4190: 4172: 4171: 4170: 4169: 4168: 4167: 4166: 4165: 4145: 4144: 4143: 4142: 4141: 4140: 4139: 4138: 4125: 4124: 4123: 4122: 4121: 4120: 4119: 4118: 4117: 4116: 4104: 4103: 4102: 4101: 4100: 4099: 4098: 4097: 4096: 4095: 4084:intuitionistic 4067: 4066: 4065: 4064: 4063: 4062: 4061: 4060: 4059: 4058: 4038: 4037: 4036: 4035: 4034: 4033: 4032: 4031: 4030: 4029: 4018:logical axioms 4014:special axioms 3997: 3996: 3995: 3994: 3993: 3992: 3991: 3990: 3989: 3988: 3972: 3971: 3970: 3969: 3968: 3967: 3966: 3965: 3955: 3954: 3953: 3952: 3951: 3950: 3913: 3909: 3903: 3902: 3901: 3900: 3899: 3898: 3897: 3896: 3895: 3894: 3871: 3870: 3869: 3868: 3867: 3866: 3865: 3864: 3842: 3841: 3840: 3839: 3838: 3837: 3813: 3812: 3811: 3810: 3792: 3791: 3748: 3747: 3746: 3745: 3700: 3699: 3698: 3697: 3696: 3695: 3694: 3693: 3692: 3691: 3690: 3689: 3688: 3687: 3686: 3685: 3649: 3613: 3612: 3611: 3610: 3609: 3608: 3607: 3606: 3605: 3604: 3603: 3602: 3576: 3575: 3574: 3573: 3572: 3571: 3570: 3569: 3568: 3567: 3566: 3565: 3528: 3527: 3526: 3525: 3524: 3523: 3522: 3521: 3509:platonism.) -- 3487: 3486: 3485: 3484: 3483: 3482: 3454: 3453: 3452: 3451: 3433: 3432: 3349: 3348: 3347: 3346: 3283: 3282: 3281: 3280: 3279: 3278: 3255: 3254: 3253: 3252: 3251: 3250: 3249: 3248: 3247: 3246: 3245: 3244: 3243: 3242: 3241: 3240: 3239: 3238: 3237: 3236: 3235: 3234: 3233: 3232: 3231: 3187: 3141: 3137: 3125: 3122: 3119: 3116: 3113: 3110: 3107: 3104: 3101: 3098: 3095: 3092: 3089: 3086: 3083: 3080: 3077: 3074: 3071: 3068: 3065: 3062: 3059: 3056: 3053: 3050: 3047: 3044: 3041: 3030: 2996: 2989: 2985: 2977: 2970: 2928: 2880: 2821: 2820: 2819: 2818: 2803: 2754: 2751: 2727: 2724: 2723: 2722: 2688: 2685: 2667: 2664: 2663: 2662: 2661: 2660: 2642: 2641: 2596: 2593: 2592: 2591: 2590: 2589: 2559: 2558: 2540:symbolic logic 2528: 2513:symbolic logic 2498:Stevenmitchell 2481:symbolic logic 2461:symbolic logic 2456: 2450: 2439:69.228.171.150 2425: 2424: 2421: 2418: 2410: 2407: 2406: 2405: 2404: 2403: 2386: 2385: 2352: 2349: 2348: 2347: 2332: 2331: 2330: 2329: 2305: 2304: 2303: 2302: 2288: 2267: 2266: 2258: 2255: 2254: 2253: 2252: 2251: 2250: 2249: 2222: 2221: 2220: 2219: 2218: 2217: 2205: 2204: 2203: 2202: 2157: 2156: 2137: 2136: 2135: 2134: 2133: 2132: 2131: 2130: 2129: 2128: 2065: 2024: 2023: 2022: 2021: 2015: 2014: 2013: 2012: 2005: 2004: 1940: 1937: 1917:Symbolic logic 1912:Category:Logic 1904:Symbolic logic 1893:Symbolic logic 1877: 1876: 1875: 1874: 1873: 1872: 1871: 1870: 1853: 1812: 1791: 1790: 1789: 1786: 1758: 1746:on Mathworld. 1736:symbolic logic 1692: 1691: 1684: 1683: 1676: 1675: 1668: 1667: 1660: 1659: 1652: 1651: 1644: 1643: 1636: 1635: 1628: 1627: 1620: 1619: 1578: 1577: 1576: 1575: 1561: 1542:symbolic logic 1521: 1500: 1483: 1466: 1447:Oddly enough, 1445: 1427: 1410: 1368: 1367: 1357:Stevenmitchell 1340:symbolic logic 1320:symbolic logic 1314: 1311:Symbolic logic 1307: 1306: 1305: 1304: 1303: 1272: 1271: 1270: 1178: 1173: 1158: 1157: 1118: 1115: 1076: 1073: 1072: 1071: 1029: 1027: 1026: 1003: 1002: 977: 976: 975: 939: 905: 902: 901: 900: 899: 898: 883: 827: 824: 823: 822: 781: 778: 777: 776: 775: 774: 773: 772: 771: 770: 769: 768: 740: 726: 690: 665: 664: 663: 662: 571: 570: 569: 568: 567: 566: 548: 547: 546: 545: 524:A-class review 517: 516: 515: 514: 507: 506: 471: 468: 460: 459: 432: 420: 419: 414: 413: 411: 410: 399: 386: 384: 383: 371: 326: 324: 317: 316: 313: 312: 309: 308: 305: 304: 299: 289: 288: 286: 284: 278: 277: 269: 268: 261:Mid-importance 257: 251: 250: 248: 216: 215: 199: 187: 186: 184:Mid‑importance 172: 160: 159: 156: 155: 144: 138: 137: 135: 118:the discussion 105: 104: 88: 76: 75: 67: 55: 54: 48: 26: 13: 10: 9: 6: 4: 3: 2: 5471: 5460: 5457: 5455: 5452: 5450: 5447: 5445: 5442: 5440: 5437: 5435: 5432: 5430: 5427: 5425: 5422: 5420: 5417: 5415: 5412: 5410: 5407: 5405: 5402: 5400: 5397: 5395: 5392: 5391: 5389: 5382: 5381: 5377: 5373: 5369: 5361: 5355: 5352: 5347: 5341: 5339: 5336: 5331: 5325: 5321: 5320: 5319: 5315: 5311: 5306: 5302: 5298: 5297: 5296: 5295: 5291: 5287: 5278: 5274: 5270: 5266: 5262: 5258:formal logic 5256: 5255: 5254: 5253: 5249: 5245: 5242: 5238: 5231: 5223: 5221: 5213: 5208: 5205: 5203: 5198: 5189: 5187: 5186: 5182: 5178: 5168: 5163: 5159: 5152: 5151: 5150: 5149: 5145: 5141: 5134: 5127: 5122: 5118: 5112: 5111: 5110: 5109: 5105: 5101: 5090: 5088: 5087: 5083: 5079: 5072: 5068: 5064: 5060: 5057: 5056: 5055: 5054: 5049: 5045: 5036: 5032: 5028: 5024: 5020: 5019: 5018: 5017: 5013: 5009: 5002: 4998: 4995: 4987: 4986: 4985: 4983: 4967: 4962: 4959: 4955: 4949: 4945: 4941: 4937: 4936: 4935: 4934: 4930: 4926: 4917: 4913: 4909: 4905: 4904: 4903: 4902: 4899: 4895: 4891: 4886: 4885: 4884: 4883: 4879: 4875: 4865: 4861: 4857: 4853: 4852: 4851: 4850: 4846: 4842: 4830: 4827: 4824: 4823: 4819: 4816: 4814:A is possible 4813: 4812: 4808: 4805: 4802: 4801: 4797: 4794: 4791: 4790: 4786: 4783: 4780: 4779: 4775: 4772: 4769: 4768: 4765: 4759: 4757: 4756: 4752: 4748: 4711: 4707: 4703: 4698: 4697: 4696: 4692: 4688: 4684: 4679: 4678: 4677: 4673: 4669: 4664: 4663: 4662: 4658: 4654: 4648: 4642: 4638: 4637: 4636: 4632: 4628: 4624: 4623: 4622: 4618: 4614: 4610: 4606: 4599: 4595: 4594: 4593: 4589: 4585: 4580: 4579: 4578: 4574: 4570: 4566: 4562: 4555: 4553: 4549: 4545: 4541: 4538: 4518: 4514: 4507: 4503: 4502: 4501: 4500: 4499: 4498: 4497: 4496: 4495: 4494: 4493: 4492: 4491: 4490: 4489: 4488: 4487: 4486: 4485: 4470: 4466: 4462: 4459: 4458: 4457: 4456: 4455: 4454: 4453: 4452: 4451: 4450: 4449: 4448: 4434: 4433: 4432: 4431: 4430: 4429: 4428: 4427: 4426: 4425: 4424: 4423: 4408: 4407: 4406: 4405: 4404: 4403: 4402: 4401: 4400: 4399: 4398: 4397: 4383: 4379: 4375: 4374: 4373: 4372: 4371: 4370: 4369: 4368: 4367: 4366: 4365: 4364: 4345: 4341: 4337: 4333: 4332: 4331: 4330: 4329: 4328: 4327: 4326: 4325: 4324: 4323: 4322: 4321: 4320: 4319: 4318: 4317: 4316: 4299: 4295: 4291: 4286: 4281: 4278: 4274: 4273: 4272: 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2768: 2764: 2759: 2752: 2750: 2748: 2744: 2740: 2736: 2725: 2721: 2717: 2713: 2709: 2705: 2704: 2703: 2702: 2698: 2694: 2686: 2684: 2683: 2679: 2675: 2671: 2665: 2659: 2655: 2651: 2646: 2645: 2644: 2643: 2640: 2635: 2631: 2625: 2621: 2620: 2619: 2618: 2614: 2613: 2607: 2606: 2594: 2588: 2584: 2580: 2576: 2575: 2574: 2570: 2566: 2561: 2560: 2557: 2553: 2549: 2545: 2541: 2537: 2533: 2529: 2527: 2523: 2519: 2514: 2510: 2509: 2508: 2507: 2503: 2499: 2495: 2491: 2486: 2482: 2478: 2474: 2470: 2466: 2462: 2455: 2451: 2449: 2448: 2444: 2440: 2436: 2435: 2430: 2429: 2422: 2419: 2416: 2415: 2414: 2408: 2402: 2398: 2394: 2390: 2389: 2388: 2387: 2384: 2380: 2376: 2372: 2368: 2364: 2363: 2362: 2359: 2350: 2346: 2342: 2338: 2337:70.90.174.101 2334: 2333: 2328: 2323: 2319: 2313: 2309: 2308: 2307: 2306: 2301: 2297: 2293: 2289: 2285: 2284: 2283: 2282: 2281: 2280: 2276: 2272: 2271:70.90.174.101 2264: 2263: 2262: 2256: 2248: 2243: 2239: 2232: 2228: 2227: 2226: 2225: 2224: 2223: 2215: 2211: 2210: 2209: 2208: 2207: 2206: 2201: 2198: 2193: 2189: 2185: 2181: 2180:general logic 2177: 2176: 2175: 2170: 2166: 2159: 2158: 2153: 2152: 2151: 2150: 2146: 2142: 2127: 2122: 2118: 2112: 2111: 2110: 2107: 2102: 2098: 2095: 2094: 2093: 2090: 2085: 2081: 2077: 2072: 2066: 2064: 2061: 2056: 2052: 2046: 2045: 2044: 2039: 2035: 2028: 2027: 2026: 2025: 2019: 2018: 2017: 2016: 2009: 2008: 2007: 2006: 2003: 2000: 1995: 1990: 1986: 1982: 1977: 1973: 1972: 1971: 1970: 1967: 1962: 1958: 1954: 1950: 1946: 1938: 1936: 1935: 1932: 1929: 1925: 1920: 1918: 1913: 1909: 1905: 1900: 1898: 1894: 1889: 1886: 1881: 1869: 1864: 1860: 1854: 1852: 1847: 1843: 1836: 1835: 1834: 1830: 1826: 1821: 1817: 1813: 1810: 1806: 1805: 1804: 1801: 1796: 1792: 1787: 1784: 1783: 1781: 1780: 1779: 1776: 1771: 1767: 1763: 1759: 1757: 1754: 1749: 1745: 1741: 1737: 1733: 1732: 1731: 1730: 1725: 1721: 1714: 1709: 1708: 1704: 1700: 1695: 1689: 1688: 1687: 1681: 1680: 1679: 1673: 1672: 1671: 1665: 1664: 1663: 1657: 1656: 1655: 1649: 1648: 1647: 1641: 1640: 1639: 1633: 1632: 1631: 1625: 1624: 1623: 1617: 1616: 1615: 1612: 1611: 1607: 1603: 1597: 1596: 1592: 1588: 1583: 1574: 1570: 1566: 1562: 1560: 1556: 1552: 1548: 1543: 1539: 1538: 1537: 1534: 1529: 1525: 1522: 1520: 1517: 1512: 1508: 1505: 1504:Rudolf Carnap 1501: 1499: 1496: 1491: 1487: 1484: 1482: 1479: 1474: 1470: 1467: 1465: 1462: 1457: 1453: 1450: 1446: 1444: 1441: 1436: 1432: 1428: 1426: 1423: 1418: 1414: 1411: 1409: 1406: 1401: 1397: 1393: 1390: 1387: 1384: 1383: 1382: 1381: 1378: 1373: 1366: 1362: 1358: 1354: 1350: 1345: 1341: 1337: 1333: 1329: 1325: 1321: 1317: 1316: 1312: 1308: 1302: 1298: 1294: 1290: 1288: 1284: 1280: 1273: 1269: 1264: 1260: 1254: 1250: 1246: 1242: 1241: 1240: 1236: 1232: 1228: 1224: 1221:, we link to 1220: 1216: 1215: 1214: 1209: 1205: 1199: 1195: 1194: 1193: 1192: 1188: 1184: 1177: 1174: 1172: 1171: 1167: 1163: 1156: 1151: 1147: 1140: 1139: 1138: 1137: 1133: 1129: 1123: 1116: 1114: 1113: 1109: 1105: 1099: 1098: 1093: 1089: 1083: 1075:Early history 1074: 1070: 1065: 1061: 1055: 1051: 1050: 1049: 1047: 1043: 1039: 1035: 1025: 1020: 1016: 1009: 1005: 1004: 1001: 997: 993: 989: 986: 982: 978: 974: 970: 966: 962: 958: 957: 956: 952: 948: 944: 940: 937: 933: 932: 931: 929: 925: 921: 917: 911: 903: 897: 893: 889: 884: 880: 879: 878: 873: 869: 862: 861: 860: 859: 855: 851: 845: 844: 840: 836: 832: 821: 816: 812: 806: 805:Das Kontinuum 802: 798: 797: 796: 795: 792: 788: 785: 779: 767: 764: 759: 758: 757: 752: 748: 741: 737: 736: 735: 732: 727: 724: 719: 718: 717: 712: 708: 701: 700: 699: 696: 691: 689: 685: 684: 683: 682: 677: 673: 661: 658: 654: 649: 645: 644: 643: 638: 634: 625: 618: 617: 616: 615: 610: 606: 600: 596: 592: 590: 586: 582: 578: 565: 562: 558: 554: 553: 552: 551: 550: 549: 544: 539: 535: 529: 525: 521: 520: 519: 518: 511: 510: 509: 508: 505: 501: 497: 492: 491: 490: 489: 484: 480: 469: 467: 466: 457: 456:its talk page 452: 447: 443: 439: 438: 433: 426: 425: 408: 406: 405: 400: 397: 395: 394: 389: 388: 385: 381: 377: 376: 373: 370: 367: 364: 361: 358: 355: 352: 349: 346: 342: 340: 336: 325: 323: 322: 303: 295: 291: 290: 287: 285: 280: 279: 274: 270: 266: 262: 256: 253: 252: 249: 232: 231: 226: 222: 221: 213: 202: 200: 197: 193: 192: 188: 181: 176: 173: 170: 166: 153: 149: 143: 140: 139: 136: 119: 115: 111: 110: 102: 96: 91: 89: 86: 82: 81: 77: 71: 68: 65: 61: 56: 52: 46: 38: 37: 27: 23: 18: 17: 5365: 5282: 5234:distinction 5227: 5219: 5214:- 5 Oct 2006 5193: 5171: 5130: 5094: 5078:Rick Norwood 5075: 5039: 5003: 4999: 4991: 4979: 4961: 4953: 4922: 4868: 4836: 4806:Linear Logic 4763: 4744: 4702:96.231.153.5 4682: 4646: 4608: 4601: 4597: 4557: 4516: 4509: 4483: 4381: 4377: 4277:WP:IMPARTIAL 4198:96.231.153.5 4193: 4185: 4134: 3854:128.8.140.59 3822: 3775: 3771: 3751: 3749: 3705:Rick Norwood 3701: 3656: 3652: 3629: 3467: 3463: 3393: 3391: 3368: 3350: 3317: 3300: 3286: 3284: 3258: 3192:96.231.153.5 3171: 3167: 3163: 3159: 2991: 2981: 2951: 2947: 2903:96.231.153.5 2884: 2861: 2828: 2799: 2786:128.8.140.59 2766: 2762: 2760: 2756: 2733:— Preceding 2729: 2690: 2669: 2650:Rick Norwood 2609: 2602: 2598: 2579:90.205.92.37 2565:66.127.52.47 2548:66.127.52.47 2512: 2484: 2458: 2453: 2437: 2431: 2426: 2412: 2354: 2311: 2268: 2260: 2230: 2213: 2141:Rick Norwood 2138: 2079: 2070: 2011:alluded to). 1942: 1922:I have left 1921: 1901: 1890: 1882: 1878: 1819: 1815: 1808: 1740:formal logic 1712: 1710: 1699:Rick Norwood 1696: 1693: 1685: 1677: 1669: 1661: 1653: 1645: 1637: 1629: 1621: 1613: 1602:JoesphPGrant 1598: 1587:Rick Norwood 1581: 1579: 1546: 1541: 1369: 1343: 1292: 1276: 1197: 1182: 1180: 1159: 1127: 1124: 1120: 1100: 1078: 1053: 1038:98.208.55.34 1028: 960: 942: 920:98.208.55.34 907: 846: 833: 829: 801:Hermann Weyl 791:Rick Norwood 789: 786: 783: 723:institutions 687: 666: 647: 597: 593: 581:64.89.212.40 575:— Preceding 572: 473: 463: 437:Formal logic 435: 402: 401: 391: 390: 372: 365: 359: 353: 347: 333: 327: 260: 228: 218: 148:Top-priority 147: 107: 73:Top‑priority 51:WikiProjects 34: 5345:Botterweg14 5329:Botterweg14 5236:(Q77227534) 4828:Modal Logic 4135:Conclusion: 4016:, but also 3291:Paul August 2767:sempliciter 2763:nonstandard 2536:old version 2076:QED project 1989:type theory 1744:WP:DICTDEFs 1742:are merely 1449:Jon Barwise 1117:Modal logic 1032:—Preceding 914:—Preceding 451:its history 123:Mathematics 114:mathematics 70:Mathematics 5388:Categories 5260:(Q1003009) 4954:References 4825:A is valid 4505:formalist. 3776:everything 3628:Time-like 3318:interested 2708:WP:RD/MATH 2693:Poppurrpop 2071:completely 1915:contentat 1231:EdJohnston 1084:). — Carl 992:Hans Adler 888:Hans Adler 850:Hans Adler 835:Skolemizer 763:Hans Adler 731:Hans Adler 695:Hans Adler 657:Hans Adler 655:, 2000. -- 561:Hans Adler 496:Hans Adler 440:page were 335:To-do list 236:Philosophy 225:philosophy 175:Philosophy 5310:Trovatore 5286:Psycho 79 4817:Lax Logic 4781:K knows A 4687:Trovatore 4668:Sammy1339 4653:Trovatore 4627:Sammy1339 4613:Trovatore 4584:Sammy1339 4569:Trovatore 4544:Trovatore 4382:imperfect 4378:imperfect 4290:Sammy1339 4080:classical 3940:Sammy1339 3827:Sammy1339 3781:Sammy1339 3721:Sammy1339 3675:Sammy1339 3555:Sammy1339 3511:Sammy1339 3441:Sammy1339 3351:See also 3322:Trovatore 3259:somewhere 3221:Sammy1339 3207:Trovatore 3177:Sammy1339 3145:Trovatore 3016:Sammy1339 3001:Trovatore 2957:Sammy1339 2933:Trovatore 2918:Trovatore 2889:Trovatore 2866:Sammy1339 2847:Trovatore 2832:Sammy1339 2808:Trovatore 2771:Trovatore 2712:Trovatore 2626:. — Carl 2611:chatties 2518:Trovatore 2391:Thanks.-- 2375:Gandalf61 2292:Trovatore 2080:manifesto 2051:WP:WEIGHT 1825:Trovatore 1551:Trovatore 1396:metalogic 965:Trovatore 947:Trovatore 530:. — Carl 39:is rated 5230:Wikidata 5212:CMummert 4558:Con(T)→G 4510:Con(T)→G 4194:intended 4010:formulas 2735:unsigned 2674:Tijfo098 2490:Calculus 1957:contents 1349:Calculus 1162:Dwheeler 1104:Dwheeler 1034:unsigned 943:predates 916:unsigned 624:citation 577:unsigned 4647:numbers 3800:YohanN7 3735:YohanN7 3661:YohanN7 3639:YohanN7 3541:YohanN7 3496:YohanN7 3420:on CZ. 3398:YohanN7 3394:Gödel's 3336:YohanN7 3303:YohanN7 3268:YohanN7 3140:number. 2494:Algebra 2155:logic". 1945:divides 1939:MSC2010 1353:Algebra 739:length. 363:refresh 351:history 263:on the 150:on the 41:B-class 5350:(talk) 5334:(talk) 5304:logic. 4024:, and 3912:pages. 3752:formal 3630:closed 3374:, and 3029:mean). 2885:models 2393:Gilisa 2231:define 2184:aspect 1820:should 1809:vastly 1764:to be 1297:(talk) 1187:(talk) 1132:(talk) 934:Not a 780:Cantor 646:Thank 442:merged 404:Wikify 393:Expand 47:scale. 5265:Daask 5244:Daask 5232:item 4925:BostX 4567:). -- 3172:might 3168:every 3164:every 3160:would 2800:about 2604:TyrS 1955:etc. 1931:Adler 1762:logic 1713:might 444:into 357:watch 302:Logic 180:Logic 28:This 5376:talk 5372:Bumv 5314:talk 5290:talk 5269:talk 5248:talk 5181:talk 5162:talk 5144:talk 5121:talk 5104:talk 5082:talk 5067:talk 5048:talk 5031:talk 5012:talk 4944:talk 4929:talk 4912:talk 4894:talk 4890:Dmcq 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