User:Mkehrt/llintro

Intuition and Motivation
Traditional logics such as propositional logic are useful for reasoning about the truth of propositions. In such logics, one can talk about the truth of propositions such as "$$A$$ and $$B$$", "$$A$$ or $$B$$" or "$$A$$ implies $$B$$" for some atomic propositions $$A$$ and $$B$$. For example, in most such logics, one can easily prove "$$A$$" from "$$A$$ and $$B$$"

However, such logics are poor tools for reasoning about the way resources, such as physical things, behave. For example, consider two atomic propostions, $$cream$$ (which we consider to mean "we have some cream") and $$butter$$ (which we consider to mean "we have some butter"). Now, imagine we have do actually have some cream which we wish to churn into butter. In propositional logic, one could naively model with as the following two propositions being true:

$$cream$$

and

$$butter \rightarrow cream$$, where $$\rightarrow$$ is read as "implies."

We could now use both of these propositions to derive $$butter$$. However, $$cream$$ is still true. Now we have both cream and butter, which is not at all how actual cream behaves.

The first step in fixing this problem is to realize that

XXXXXXX LL is "substructural" in that...