# Tagged Questions

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### What's the difference between a logic, an internal logic (language) of a category, an internal logic of a topos and a type theory?

maybe this question doesn't make sense at all. I don't know exactly the meaning of all these concepts, except the internal language of a topos (and searching on the literature is not helping at all). ...
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### Relation between existential and universal quantificator in category theory

Let $\mathscr C$ be a cartesian (i.e. with finite limits) category with subobject classifier $\Omega$ and generic subobject $\tau:I\to\Omega$ (here $I$ denote the terminal object). Let $f:X\to Y$ and ...
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### Relations between Theories and Categories

I'm just toying around with some thoughts, trying to grock some concepts: It seems that every formal theory induces a locally small category via interpretations: its objects are structures that ...
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### Is there a formal way to describe classical logic as a reflective subcategory of constructive logic?

Working informally, we can take any proof $P$ in constructive (or intuitionistic) logic and turn it into a classical proof $cP$ by 'copying' it, since all the rules of constructive logic reappear in ...
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### Is there a phrase to describe those objects of $\mathbf{C}$ that can be expressed as quotients of the algebra freely generated by $X$?

Let $\mathbf{C}$ denote the category of models of an algebraic theory in $\mathbf{Set}.$ Now suppose $X$ is an object of $\mathbf{Set}$. Is there a traditional phrase used to describe those objects of ...
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### Simplifying a categorical proof of constructive dilemma

In axiomatic propositional calculus the following axiom schema captures constructive dilemma: $\newcommand{\lif}{\supset} \renewcommand{\land}{\&}$ (a \lif c) \lif ((b \lif c) ...
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### Arrows-only implication & disjunction in $\mathbf{Set}.$

Just before the truth-arrows in a topos subsection of Goldblatt's "Topoi: A Categorial Analysis of logic," descriptions of the truth functions $\Rightarrow$ and $\smallsmile$ are given in ...
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### Is there any relevance between Boolean Algebras and Fields?

In some sense Boolean Algebras and Fields have same operators and constants. In both structures there are operators addition ($+$ , $\vee$), multiplication ($\times$ , $\wedge$), inverse with respect ...
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### Is there a first order theory for equivalences classes?

Question will be a bit naive, so please, be kind. Consider a first order theory, $\Gamma$ . Let $\mathcal{M}$ be the category of models for $\Gamma$. Consider $\sim$ an equivalence relation on ...
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### Resources for Polyadic and/or Cylindric Algebra

I'm looking to learn a little bit about polyadic and cylindric algebras, as part of an investigation into algebraic approaches to logic. The only "text" that I can find for polyadic algebra is ...
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Lawvere's Sets for Mathematics defines a separator object $S$ for morphisms $f_1, f_2: X \rightarrow Y$ of the Set category as a logic statement $s$ $s$: $\forall x \left[ S \xrightarrow{x} X ... 3answers 135 views ### Some reference for categorical logic? By "categorical logic" I mean category-theoretical models of logic. In particular, I am more interested in models of intuitionistic predicate logic with conjunction, disjunction, implication and ... 2answers 512 views ### Is maths = set theory + logic? [closed] It seems to me that most branches of mathematics are just an application of set theory and the rules of logic. You just definite particular types of sets and study their properties. For example, in ... 1answer 104 views ### What do identities mean in$\mathrm{Set}^\mathrm{op}$? Since$\mathrm{Set}$has finite coproducts, thus we may consider models of equational theories in the opposite category$\mathrm{Set}^\mathrm{op}$. The result is basically that function symbols$f : ...
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In my opinion, a cool definition of "algebraic signature" is as follows: An algebraic signature on the sort symbols $\mathcal{X} = \{X_0,...,X_{n-1}\}$ is precisely a quiver whose underlying set ...
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### What is the dual of implication?

You may divide Intuitionistic Propositional Logic into the negative and positive fragments. The negative fragment includes truth, conjunction, and implication while the positive fragment includes ...
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### Can we capture all domains of discouse in the predicate logic within categorical logic?

In the construction of the bounded quantifiers via adjoints in the fibered category of subsets over a set (see e.g. here on Wikipedia), is there any restriction on the sets - specifically regarding ...
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### Advice regarding best-practice mathematics / categorial logic.

A good heuristic is: If it doesn't cost anything, generalize. In particular, if we have a theorem, and a proof thereof, then we ought to look for a maximal generalization of this theorem, ...
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### Beyond Goedel incompleteness and lack of soundness/completeness of higher-order logics

As I understand that there are at least two fundamental limits of the development of the mathematics: 1) Goedel incompeleteness theorems (or more clearly Church thesis) effectively says that there ...