Chaining is the logic of Eberban, and Arity Mismatch Resolution is its linchpin. We’ve decided that the latter is foundational enough to come first in its own chapter.
The function of this concept lies in its name. Arity mismatches occur all the time. In this chapter, we’ll look at how they occur in sentences and how they are resolved.
🪶 Jargon: Arity mismatch
Given two side-by-side words, an arity mismatch occurs when the expected arity of the left is less than the actual arity of the right.
Occurring in sentences
In Eberban, sentences are assertions: statements of what the speaker believes to be true. Assertions aren’t necessarily true, in fact, each assertion evaluates to trivalent truth.
🪶 Jargon: Trivalent truth
We define trivalent truth to be the set {(proveably) true, (proveably) false, unknown/unproven}.
Similar to a full stop in English, an Eberban sentence is marked by “pa” at the beginning.
“pa” is a particle that expects a 0-ary predicate. This is because only propositions can be asserted.
✍️ Examples:
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Loading...: - pa e te dena: “In the context, it is Monday.”
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Loading...: - pa e tia gare: “In the context, it is June.”
In future examples, we’ll omit the context unless it is the focus.
These examples work as expected. Let’s use some other predicates:
✍️ Examples:
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Loading...: - pa mian: ❌ arity mismatch: 0 < 1
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Loading...: - pa gli: ❌ arity mismatch: 0 < 2
These don’t. It’s analogous to trying to force two jigsaw pieces together.
Time to resolve the mismatch!
The process
Arity Mismatch Resolution only operates on two side-by-side words. Let’s call these words L and R for left and right respectively.
Beginning with arity of L < arity of R:
- Choose a place of R.
- Replace it with an existential argument.
- Go back to step 1 until the arities of L and R match.
We’ll learn which place gets chosen when we learn Chaining.
🪶 Jargon: Existential arguments
An existential argument is an argument that is created by the process of Arity Mismatch Resolution to replace a place of a predicate.
Resolved sentences
Now that we know the process, let’s see what our sentences look like:
The words haven’t changed at all! That’s because Arity Mismatch Resolution happens automatically behind the scenes. This helps keep Eberban sentences concise, letting us focus on expressing meaning. Here is the translation of our sentences:
✍️ Examples:
- pa mian: There exists some physical entity+ e which is a cat.
- pa gli: There exists some physical entity+ e and some proposition A, such that e is happy that A is true.