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I just noticed there is a similarity between logic operations on propositions and the operations of set theory. It seems:

$$\begin{array}{llll} \textrm{disjunction} & (-)\vee (-)& \textrm{corresponds to union}& (-)\cup (-)\\ \textrm{conjunction} & (-) \wedge (-)& \textrm{correspons to intersection} & (-)\cap (-)\\ \textrm{negation} & \sim (-) & \textrm{correspons to taking complements} & c(-), \end{array} $$ and I conjecture:

$$\begin{array}{lll} \textrm{conditional} & (-)\rightarrow (-) & \textrm{corresponds to inclusion} & \subset\\ \textrm{biconditional} & (-)\leftrightarrow (-) & \textrm{corresponds to equality} & = \end{array}$$

How far does it go? I believe there is some kind of functor between some category whose objects are propositions and the category of sets, is that right?

Thanks

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  • $\begingroup$ The category of sets deals with functions from one set to another. What will be the arrows in your proposed category where the objects are propositions? Summary: Yes, there is a parallel here, but you should not bring in categories. $\endgroup$ – GEdgar Mar 21 '19 at 12:57
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    $\begingroup$ For conenctio, see Boolean algebra as well as The Mathematics of Boolean Algebra. $\endgroup$ – Mauro ALLEGRANZA Mar 21 '19 at 13:11
  • $\begingroup$ See also The Algebra of Logic Tradition for historical development. $\endgroup$ – Mauro ALLEGRANZA Mar 21 '19 at 13:12
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    $\begingroup$ The parallel between $\to$ and $\subseteq$ is not quite direct. Note that $\to$ combines two propositions into a proposition that you can use with further logical symbols, whereas $\subseteq$ combines two set expressions into a proposition. So whereas you can say $A\land(B\to C)$ in logic, you cannot meaningfully say $A\cap(B\subseteq C)$ in set algebra. (Similarly for $\leftrightarrow$ versus $=$). $\endgroup$ – hmakholm left over Monica Mar 21 '19 at 15:54
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    $\begingroup$ The simplest connection is to identify each set $S$ with the proposition $a\in S$ for some constant $a$, or each proposition $p$ with the set of conditions under which it's true. There's also another natural concept for analysing propositions. $\endgroup$ – J.G. Mar 21 '19 at 15:55
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"How far does it go ?" : as far as it can get !

Here's a general idea about how set theory is a semantics for classical propositional logic (note that if we change the formal system we're looking at without changing our assumptions on sets, for instance if we're studying intuitionistic logic from a classical point of view, then we have to take another semantics, in this specific case, topological spaces can be appropriate) :

Suppose you have a set of propositional variables $V$, a "global" set $E$, and a function $[-] :V\to \mathcal{P}(E)$. Then you can build a function that goes from the set $\mathrm{Form}$ of formulas to $\mathcal{P}(E)$ by expanding $[-]$ according to the rules you displayed : if $\varphi, \psi$ are formulas and you already defined $[\varphi]$ and $[\psi]$, then define $[\varphi \land \psi] = [\varphi]\cap [\psi]$, similarly for $\lor, \neg$, and define $[\varphi\implies \psi]$ as $\{x\in E\mid x\in [\varphi]\implies x\in[\psi]\}$.

These rules allow you to define $[\varphi]$ for any formula $\varphi$ by induction, going from the variables and gaining complexity.

Then you can prove the following things : if $\varphi$ is a theorem of classical logic, then $[\varphi] = E$, which tells you that the set-operations behave according to the logical ones, but also you can prove : if for any $E$ and any $[-] :V\to \mathcal{P}(E)$, $[\varphi] = E$, then $\varphi$ is a theorem of classical logic ! This tells you that actually the logical operations behave just like set-theoretic operations as well.

There's actually a lot more you can say about this sort of thing (for instance : what happens if you add quantifiers ? Or in another direction what happens if we replace $\mathcal{P}(E)$ by some other type of object ? If we completely change the type of object, what kind of logic do we get ? etc. etc.)

If you absolutely want to use the words "functor" and "category" you can, but at this level they're not the most relevant thing.

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Yes, there is an abstract isomorphism here and the likeness of the symbols $\lor$ and $\cup$, as well as that of $\land$ and $\cap$ is of course no accident!

Also, if you use the formal definition of the set operators, we see the connection there as well:

Union:

$\forall A \ \forall B \ \forall x \ (x \in A \color{red}\cup B \leftrightarrow (x \in A \color{red}\lor x \in B))$

Intersection:

$\forall A \ \forall B \ \forall x \ (x \in A \color{red}\cap B \leftrightarrow (x \in A \color{red}\land x \in B))$

Complement:

$\forall A \ \forall B \ \forall x \ (x \in A\color{red}' \leftrightarrow \color{red}\neg x \in A)$

And your conjecture is right in that we also have:

Inclusion:

$\forall A \ \forall B \ (A\color{red} \subseteq B \leftrightarrow \forall x (x \in A \color{red} \rightarrow x \in B))$

Equality:

$\forall A \ \forall B \ (A\color{red} = B \leftrightarrow \forall x (x \in A \color{red} \leftrightarrow x \in B))$

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The connection is math is a logical science. It's built on a foundation of axioms and definitions, from which ideally, all statements can be proven, or disproven ( though some are undecideable sadly). You can have logical properties with set operations, Sets equipped with operations, having certain properties applied on the set, are the basis for: magmas, monoids, loops, semigroups, quasigroups, groups, abelian groups, rings, commutative rings, and fields, just to name a few buzzwords. Mathematical logic, can be defined in terms of other more basic logics.

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