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Recently I read the chapter "Doctrines in Categorical Logic" by Kock, and Reyes in the Handbook of Mathematical Logic. And I was quite impressed with the entire chapter. However it is very short, and considering that this copy was published in 1977, possibly a bit out of date.

My curiosity has been sparked (especially given the possibilities described in the aforementioned chapter) and would like to know of some more modern texts, and articles written in this same style (that is to say coming from a logic point of view with a strong emphasis on analogies with normal mathematical logic.)

However, that being said, I am stubborn as hell, and game for anything.

So, Recommendations?

Side Note: More interested in the preservation of structure, and the production of models than with any sort of proposed foundational paradigm

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Community wiki? –  Ed Dean Jan 25 '11 at 4:59
    
Sure, that works. –  Michael Blackmon Jan 25 '11 at 5:10
    
Decided to not select an answer because so many of them are really good, and I'm having a great time going over some of the texts recommended thus far. –  Michael Blackmon Jan 27 '11 at 9:31
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8 Answers

Might I recommend Sheaves in Geometry and Logic by MacLane and Moerdijk. To quote bits from the blurb:

Sheaves also appear in logic as carriers for models of set theory as as for the semantics of other types of logic.

The applications to axiomatic set theory and the use in forcing ...are then described.

...the construction of topoi related to geometric languages and logic.

(Edit: Ed Dean beat me to it, but only just)

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Well, for the record, there is also my Practical Foundations of Mathematics (CUP, 1999).

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For those like the OP whose curiosity is sparked by doctrines of categorical logic, I think your book provides quite a lot of thought-provoking material. Another thing I like about it is the wealth of eye-opening exercises. My own experience was that the book required repeated and persistent efforts to get 'underneath', so to speak, but if the OP is "game for anything and stubborn as hell" as he says, then he could surely get a lot out of it, and he may find himself returning to it again and again. –  Todd Trimble Mar 19 '11 at 19:23
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I still understand only a fraction of this book, but it's already one of the most well-thumbed books in my library. Every time I come to it, I find something new there. –  Neel Krishnaswami Mar 20 '11 at 17:05
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Just as technical texts, Acessible Categories by Makkai and Paré and Locally Presentable and Accessible Categories by Adamek and Rosicky are extremely useful even for non-logicians (they are a natural extension of the material in SGA4.1.i on colimits of functors indexed by "ensembles ordonné grand devant $\alpha$", which are now called $\alpha$-filtered colimits), but additionally, they cover a categorical approach to model theory that is also supposed to be pretty interesting (although I haven't read those parts of the books, I've heard good things from a number of people, including François Dorais (indirectly), who is one of our benevolent moderators).

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Introduction to Higher Order Categorical Logic by Lambek and Scott might fit the bill.

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I'd like to add Carsten Butz' "Regular categories and regular logic". It is available online in the BRICS lecture series, and is very accessible. Though perhaps a bit too basic at times for readers with some background knowledge, it is very suitable for a first introduction before jumping into texts on toposes.

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For a introduction:

1) Notes on Logic and Set theory (cap. 1, cap 3) P.T. Johnstone.

2) Locally Presentable And Accessible Categories by J Adamek J Rosicky (Cap. 3 & cap. 5)

FOr a comprehensive view:

1) Sketches of an Elephant: A Topos Theory Compendium (VOl 2, cap D1)

3) B. Jacobs, Categorical Logic and Type Theory, Studies in Logic and the Foundations of Mathematics 141

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My answer here has a number of good references: Resources for learning practical category theory

I don't recommend Goldblatt (Here is an article by Colin Mclarty that elaborates on why http://www.jstor.org/pss/687825).

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Avaliable online for free: bjps.oxfordjournals.org/content/41/3/351.full.pdf –  darij grinberg Jan 25 '11 at 9:41
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http://en.wikipedia.org/wiki/Categorical_logic has more refs. That's a sufficiently obvious place to look that maybe someone can move this "answer" to a comment.

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