# Overview and introduction to strong logics

Is there a nice and simple paper which summarizes the definitions and properties of strong logics? When I say strong logics I mean something like stationary logic, or Magidor-Malitz quantifier, $\cal L_{\kappa,\lambda}$, etc.

What I am looking for is a paper without many proofs (although preferably with some proofs, just to get the idea) which gives out the definitions of the various extensions of first-order logic, and outlines their properties (compactness, completeness) and differences.

Of course a combination of several short papers is also welcomed, but I really wish to avoid (for now) the long and technical expositions on each logic.

I want to say out that at this moment I am particularly interested in the three logics mentioned above, the stationary logic, MM-quantifier, and $\cal L_{\kappa,\lambda}$ logics. Other types of strong logics are very welcomed, but those three are currently the main points of interest.

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Compactness is generally sacrificed when passing to infinitary logics: it's a fun exercise to find an (infinite) list of propositions (in, say, infinitary propositional logic) whose finite fragments are consistent while the list as a whole is inconsistent. (Hint: pigeonhole principle.) – Zhen Lin Nov 20 '12 at 0:33
Zhen, I know that. However one may still retain partial compactness, or weak-compactness. Especially if one does not fear large cardinals. :-) – Asaf Karagila Nov 20 '12 at 0:49

Peter Koellner's Strong Logics of First and Second Order seems to be the sort of article you might be interested in reading. Here's a link to an abstract to the article: Bull. Symbolic Logic Volume 16, Issue 1 (2010), 1-36..

If I recall, there are abundant references cited in the article; if this article doesn't suit your needs, one of the cited references may be more appropriate.

EDIT - the logics in question are all mentioned on the first page of the following reference:

See Part B and/or Chapter IV: The Quantifier "There Exist Uncountably Many" and Some of Its Relatives, (edited?) by M. Kaufmann], openly accessible @ ProjectEuclid. Alternatively, consider looking into the source of Chapter IV: J. Barwise, S. Feferman, eds., Model-Theoretic Logics (New York: Springer-Verlag, 1985), 123-176.

(Here is a permanent link to the article/chapter @ProjectEuclid.)

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Wonderful! I will take a look tomorrow and let you know. Thank you! – Asaf Karagila Nov 20 '12 at 0:27
@Asaf: Oh, my gosh! I didn't see that it was your question. I just saw the question. Let me know what you think. Perhaps Peter Smith can chime in too. – amWhy Nov 20 '12 at 0:29
I went over the paper, it is very interesting and I will definitely read it soon. I am particularly interested in stationary logic and the Magidor-Malitz quantifier, neither of which is mentioned in this paper. – Asaf Karagila Nov 20 '12 at 7:36
Asaf: I included an additional reference that seems to speak directly to the logics you specify. – amWhy Nov 20 '12 at 14:41
I haven't read the chapter yet, but I have copied the entire book to my computer. I am actually interested in studying a draft by my advisor and two people. One of those people wrote a very relevant chapter. Either way, I think this should give me most of what I want to know (even if I have to slightly distill it from details). Thank you very much! – Asaf Karagila Nov 21 '12 at 23:26

This paper of Vaananen should be quite useful: http://www.math.ucla.edu/~asl/bsl/1001/1001-004.ps

Other texts that you may find useful are Shelah's paper "Generalized quantifiers and compact logic" (which introduces stationary logic) and Barwise's recent book on AEC, which has some material on infinitary logic and generalized quantifiers as well.

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Thanks. I hope you are coming to my lecture on Wednesday. – Asaf Karagila Nov 24 '12 at 18:47