Talk:Categorical logic

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message: Some introductory logic texts use the term "Categorical Logic" to refer to plain, ol' Aristotelian logic, which has nothing to do with category theory (aside from the fact that category theory can be used to understand everything).  I've taught from two texts which use this terminology.  Do you suppose that the article should acknowledge this ambiguity and give readers a link to the appropriate article?  (Which article?)  Phiwum (talk) 13:36, 28 May 2009 (UTC)
thread_title:  Ambiguous title? 

message: I removed this sentence from the first paragraph of Historical perspective because no citation is provided.  Although this is a widely held opinion, it is false.  Colin McLarty, a real authority on the subject, says on the first page of the preface to his book Elementary Categories, Elementary Toposes:
:In Part III the first order topos axioms are used to define the higher order internal language. The logic is called 'topos logic' here, although 'intuitionistic logic' is often referred to in the literature. I believe that is usually, and rightly, taken to mean Brouwer's epistemology of mathematics, which is unrelated to the origin or content [emphasis mine] of topos theory. Topos logic strongly resembles formal intuitionistic logic and the two have interacted, as in .... But topos logic coincides with no intuitionistic logic studied before toposes, and this is due to real philosophical differences..."--Foobarnix (talk) 19:36, 5 July 2015 (UTC)
thread_title:  The resulting logic is formally intuitionistic 

message: ;I removed the entire paragraph below from the Historical perspective section:
:To go back historically, the major irony here is that in large-scale terms, intuitionistic logic had reappeared in mathematics, in a central place in the Bourbaki–Grothendieck program, a generation after the messy Brouwer–Hilbert controversy had ended, with Hilbert the apparent winner. Bourbaki, or more accurately Jean Dieudonné, having laid claim to the legacy of Hilbert and the Göttingen school including Emmy Noether, had revived intuitionistic logic's credibility (although Dieudonné himself found Intuitionistic Logic ludicrous), as the logic of an arbitrary topos, where classical logic was that of 'the' topos of sets. This was one consequence, certainly unanticipated, of Grothendieck's relative point of view; and not lost on Pierre Cartier, one of the broadest of the core group of French mathematicians around Bourbaki and IHES. Cartier was to give a Séminaire Bourbaki exposition of intuitionistic logic.
;Reasons for removal
:As we've seen from McLarty, intuitionist logic is not topos logic [I could give other sources]. The "had" is the wrong tense anyway. There is no such thing as "the Bourbaki-Grothendieck program" and if there were, it would have almost nothing to do with categorical logic. Grothendieck was unaware of the relations of his topos theory to logic; the relation was entirely the discovery of Lawvere. Briefly, G invented what he called a "topos" for his work in algebraic geometry; today one speaks of a "Grothendieck topos". The term "topos" usually refers to what is sometimes called an "elementary topos" (that's what Lawvere called it), a generalization of G. toposes due to Lawvere, with the aid of Tierney. A "Grothendieck topos" is not the same as an ("elementary" or "Lawvere-Tierney") "topos", and they are never confused.
:As for "Jean Dieudonné, having laid claim to the legacy of Hilbert", this is totally false.  Dieudonné worked in many fields, but never in logic. Dieudonné reviving intuitionist logic's credibility? This is totally false. Brouwer's views on logic were given a respectable interpretation by his student Arend Heyting. Andrey Kolmogorov came up with approximately the same logic from a quite different viewpoint. Today one speaks of the BHK-interpretation (Brouwer-Heyting-Kolmogorov). Again, neither Dieudonné, nor any other Bourbakiste, had anything whatever to do with this.--Foobarnix (talk) 20:15, 5 July 2015 (UTC)
thread_title:  Paragraph removed from Historical perspective section 

message: This article is OK as far as it goes.  But I think it needs to be cleaned up a bit.  I am not personally competent to do this (I tried).  But I invite someone who is more knowledgeable than me to use some version of the remarks below to improve the article.
The article should say that classically (i.e. before categorical logic), logic dealt with interpretations in the category of non-empty sets and functions., so that e.g. it could not handle semi-groups since there is an empty semi, group.  This limitation was due to an improper understanding of how to handle free variables: the categorical view shows that one must consider a term or formula or sequent not by itself but together with a list of variables that contains all the free variables of a term, formula, or sequent and possible some other variables; this list is called the context.
  
Perhaps most importantly, however, is that pre-categorical logic inherited from Frege the requirement  that variables vary over the entire universe, and consequently that equality was "universal", in the sense that t "x = y" was considered a meaningful statement, no matter what x and y are. In contrast, categorical logic is  generally "typed" , so that each variable ranges over one type only (leaving aside the possibility of subtyping), and "x = y" is an allowable statement only if x and y are of the same type. [It may be that this last part is too vague because of the omission  of subtyping].--Toploftical (talk) 14:44, 16 December 2015 (UTC)
thread_title:  Frege, semigroups, and the categorical view 

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