Federal Fluminense University, Rio das Ostras, Brazil One way to explain intuitionistic logic to a non-logician is this.
The usual truth-values are just 0 and 1, and we will change that by
decreeing that the new truth-values will be certain diagrams with
several 0s and 1s. We choose a subset D. A modal truth value
is unstable when it has a 1 immediately above a 0, for example
_{1}^{1}_{0} is unstable, and an intuitionistic truth-value on
is a stable modal truth-value on DD. Now that we have defined
our (intuitionistic) truth-values we explain to our non-logician
friend how to interpret ⊤, ⊥, ∧, ∨, → on them, and
we show that if P = _{1}^{0}_{1} then P ≠ ¬¬P = _{1}^{1}_{1}, and some other classical theorems also do not hold.
We then explain some logical axioms and rules that do hold in this
system, define intuitionistic propositional logic from them, show how
this particular case based on D generalizes, present the standard
terminology, and so on.
When we do this we are using several tricks – finding an
insightful particular case, doing things in the particular and in the
general cases in parallel using diagrams with the same shapes, lifting
proofs from the particular case to the general one –, and this
dydactical method can be defined precisely. In the terminology of [Ochs2013] this logic on subsets of ℕ
Take any ZDAG There is a system of coordinates that we can put on a ZHA – the
(
The usual way of presenting HA modalities in the literature is by
showing first some basic consequences of the axioms, then how
modalities interact with ∧, ∨, →, then theorems about how
the algebra of modalities behave; I have always found this approach
quite opaque. By using ZHAs we can explain these theorems and exhibit
conter-models for all non-theorems visually – and it turns out that
modalities on a ZHA |
It turns out that ZDAGs are categories, and ZHAs are CCCs, both
archetypal in slightly weaker ways than
Using ZToposes as our archetypal toposes we can understand how all
these entities and definitions behave by generalizing a few examples
where the diagrams are not too big. One nice example – of the The possibilities for exposing tecnicalities using archetypal cases are endless, but we will dedicate the best part of our energy in this tutorial not to them, but to something more general and more useful: how to use archetypal cases to make the literature more accesible, and to create bridges between different notations. Bibliography
[FourmanScott1979]: M.P. Fourman, D.S. Scott: [Ochs2013]: E. Ochs: |