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Category Theory Illustrated: Logic (2021)
Now let’s talk about one more seemingly unrelated topic just so we can “surprise” ourselves when we realize it’s category theory. By the way, in this chapter there will be another surprise in addition to that, so don’t fall asleep.
I don’t know about you, but I feel that the classical truth-functional interpretation of logic (althought it works and is correct in its own right) doesn’t fit well the categorical framework that we are using here: It is too “low-level”, it relies on manipulating the values of the propositions. This is another example of the categorical concept of duality - $\top$ and $\bot$ are dual to each other, which makes a lot of sense if you think about it, and also helps us remember their symbols (althought if you are like me, you’ll spent a year before you stop wondering which one is which, every time I see them). If you remember the definitions of the categorical product(or of its equivalent for orders, the meet operation) you would already know where this is going: we recognize that $A \to B$ is the upper limit of $A → B ∧ C$ and $A → B ∧ C ∧ D$ and all other imposter formulas that can be in the place of $X$ in $A ∧ X → B$.
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