I figured this would be helpful for some since recent posts about logic might seem far more complex than appropriate for a subreddit claiming to be about minimalism.

TL;DR: Paying careful attention to meaning can reveal patterns that your language can use to do more with less.

Finding abstract concepts

Any language aiming for a small word/morpheme list needs to do this at some point, and one of the best ways to do this is to examine groups of related concepts and see what the simplest categorizations are. These abstract concepts often don't have simple descriptions in natural languages, so they may counterintuitively seem more complex, but there are plenty of cases where using such concepts can simplify things overall, since often we don't need all the specificity that naturally-evolved terms give. A good example of this is toki pona, where the dictionary entries for words often have to list multiple possible interpretations.

This isn't limited to vocabulary, either, since we can also examine the effects of other productive features of natural languages, such as verb tense, and figure out how these could be applied to other things (like "noun tense", where nouns are marked for time). And when you recognize enough similarities between the meanings of verbs and nouns to make these features, you might even be able to do away with multiple lexical categories.

How intuitionistic logic (might) help

For those who aren't experts in formal logic and such (which is honestly pretty understandable), intuitionism is the position that, in order to show that something is true, one must be able to specify an exact instance of it, a "construction". It's usually expressed in a more obtuse way (with appeal to the concept of "proof" while defining what it means to prove something), but that's the practical implication. This has weird implications for logic (for instance double negation / "not not" can't generally be eliminated), which is why "intuitionistic logic" exists.

My stance on this is a little nuanced: While I don't think the intuitionistic position is correct at face value (i.e. that truth = constructibility; it definitely doesn't in English), I do think that a language which replaces the role of truth with constructibility in its semantics could make a lot of statements much shorter (especially since we can get still describe truth in those terms by double negation; going into why would take some space though). Though I haven't actually experimented with it much, I like the potential it gives as now the meanings in my language could include information not just about when something is correct, but what it takes to fully demonstrate it.

Finally an example

Let's consider the example of a dice roll, which I'll call šadzi /ɕadzi/. If I just asserted "šadzi." in a language with intuitionistic semantics, that means I claim to know not just that the thing in question is a dice roll, but that I know what the result was (since that's essential to what makes a dice roll, that's part of the meaning). I'd need to soften the claim with a modifier ("šadzihu.") to just say that it's a dice roll.