Model theory of intuistionistic logic

Question

1. The Tarskian interpretation of first order logic, $I=I_{S,\alpha}$, where $S$ is a first order structure w.r.t. the signature $\Sigma$, and $\alpha$ is a variable assignment, assign either $T$ or $F$ to every w.f.f. $\varphi$ over $\Sigma$. Is there a similar standard interpretation of intuistionistic logic?

2. The following question is only relevant assuming there is a standard interpretation of intuistionistic logic, $J$. Tarski's $I$ satisfies that $I\big(\neg(\varphi)\big)=\neg I(\varphi)$ (where $\neg T=F$, and $\neg F=T$), and that $\varphi$ is provable per Gentzen's Natural Deduction iff $I(\varphi)=T$. How does $J$ compare to $I$ in these regards?

Answer 1

It sounds as if you need a good intro to intuitionistic logic with, in particular, a discussion of (at least) the now standard Kripke semantics.

So take a look at e.g. Ch. 8 “Intuitionistic logic” of the Beginning Mathematical Logic Study Guide, which you can freely download at https://logicmatters.net/tyl

The chapter begins with a brief 8-page outline of the syntax and semantics of intuitionistic logic (including the so-called BHK interpretation and Kripke semantics), and explains why excluded middle fails on these semantic accounts. Then §8.4 suggests some further readings which will tell you a bit more.