Gated Epistemic Calculus (GEC): An Introduction, Part 1

wipWIP!!!

sStatus - about 70% done.

iIntro

eEveryone has beliefs. bBut not everyone has examined them, and those who have may not have had the tools to examine them well. hHow do you know whether your beliefs are internally consistent? wWith misinformation rampant, how can we know what to believe? aAnd how can we reduce polarization and collaborate to make the world a better place?

gGated epistemic calculus (gecGEC) is a way to assign beliefs to claims and calculate how beliefs flow into each other. cCombined with the right software, the magic then reveals itself - not only can you clearly see your beliefs, evidence, definitions, and arguments relate to one another, you can also mathematically identify hypocrisy! mMultiple people can collaborate on the same belief map to clarify ambiguities and pinpoint exactly how their beliefs differ. tThe applications are endless. yYou can use it to analyze scientific papers, come to decisions, assess candidates, and understand how your own mind works. oOn a larger scale, a "belief map wiki" or communities may provide complex yet consistent belief maps for personal modification, adoption, and analysis.

dDuring gecGEC's development iI made several philosophical and mathematical design decisions. oOne of these was working with subjective truths, not objective ones. tTwo people may have internally consistent belief systems, but due to upbringing, exposure to varying information, and other factors, arrive at different subjective truths. bBut making your individual complex belief system fully consistent is, perhaps, near impossible for the human mind with the tools we have now. tThis is the gap gecGEC is attempting to bridge.

gecGEC eEssentials

tThe lLean

gecGEC beliefs act on propositions (true-false claims). hHowever, instead of assigning true or false to the claim, you assign a number between 0 and 1, aka the lean 𝑙$l$. 1 indicates you think the claim is true, 0.5 means you don't know whether the claim is true or false (there is conflict), and 0 indicates you think the claim is false. aAs an example, take the claim "the café is worth recommending." lLet's say you think it's pretty good, so your lean is 0.75.

hHow sure are you of your answer? 0.75 seems fairly specific. hHere, gecGEC presents a knob for you to turn, the strength 𝑛𝑙$n_l$ (mathematically, 𝑛$n$ stands for the "number of trials"). tThis is a number 0 or greater that can be interpreted as "how likely am iI to change my mind about my lean," where a higher strength means you are less likely to change your lean. tThe exact strength number is subjective (the whole project is about subjective truth), but we can calibrate across people it in different ways. iI'll cover that in later parts but for now, let's just say your lean strength is a 10.

tTogether, the lean and the strength create what's known as a beta distribution. mMathematically, lean 𝑙$l$ is the mode and strength 𝑛𝑙$n_l$ is the number of trials, or in bBayesian terms, a measure of the total evidence. aA beta distribution has parameters 𝛼$\alpha$ and 𝛽$\beta$ and they are related to lean and strength in this way:

vVagueness

iImagine your friend says "sSloths are good". wWhat kind of sloths? dDo sloths have a sense of morality? wWhat does it mean for a sloth to be good? gGood for what, the ecosystem? wWhat if (hopefully not) she wants to know what sloths taste like?

tThis vagueness shows up everywhere in "simple" yes or no questions (which can be restated as claims). "iIs the policy fair?", "are his actions wrong?", "is the commute long?" tThe typical yes-no binary is often not sufficient to address your true thoughts about these questions, and gecGEC provides a potential third response - "your question is vague."

tThe real world is messy, however, and claims typically aren't perfectly clear. gGoing back to "the café is worth recommending," someone seeing the claim might have questions. wWhich cafe are we talking about? tTo who are we recommending it to? hHere gecGEC allows you to express the sense that the claim is underspecified using a term we call vagueness 𝑣$v$, in contrast to how evaluable it is, and like lean, it's a number between 0 and 1. sSo in this example you might assign something like 0.65 - the claim is pretty vague since we don't even know what cafe we're talking about.

aAlready by assigning numbers to propositions we see a benefit. tThe high vagueness means that there is an opportunity to clarify and "fix" the proposition. lLet's make it more specific - the proposition is "cCafé mMira is worth recommending." mMuch clearer, but still a bit of vagueness around the "worth recommending" part. sSo let's adjust the vagueness to 0.1. aAgain calibrating these numbers across people might be difficult, but it can be done and iI'll explain how in a future blog post.

nNote that vagueness in gecGEC is not the same as vagueness in philosophy. iIn philosophy, vagueness refers to the lack of clear boundaries between categories (e.g., where does a heap of sand become not a heap?). iIn gecGEC, vagueness is an umbrella term that includes philosophical/linguistic concepts like underdetermination, indeterminacy, context sensitivity, and ambiguity. iIn fact, in early versions of gecGEC, it was called "underspecification", but "vagueness" was chosen so it wouldn't conflict with jJosang's subjective logic (an opinion framework worth looking into), which uses 𝑢$u$ for uncertainty. "uUnderspecification" is also a mouthful.

tThe gGate

nNow, we have these two beta distributions - one for lean and one for vagueness. bBut aren't they related somehow? hHere we finally get to why gecGEC is called gGated eEpistemic cCalculus. iIt's difficult to evaluate a vague claim, so we call it less evaluable. qQuantified lean only applies to the evaluable part of a claim, and that's where we get the gate - the more vague a claim is, the more the gate closes. gecGEC beliefs are nested beta distributions, where the outer gate is the vagueness distribution and the inner gate is the lean distribution.

gecGEC rRelational pPropositions

aA fundamental part of gecGEC is propositions, and this extends to how they relate to one another. gecGEC's relational propositions determine how evidence flows from one proposition-belief pair to another.

gecGEC eExtended

aAs a modeling choice, gecGEC limits itself to beliefs on propositions. iI got great feedback from a professor that thinking isn't limited to propositions, and that a "propositional world" has been thrown around in philosophy before (there are varying amounts of similarity to logical atomism, fFregean facts, cCarnap's work, and logic as a whole). aAs such, iI got to thinking about potential ways gecGEC could integrate with other paradigms.

aA very natural extension of propositions, where true and false are the choices, are multinomial choices, such as between political candidates. tThis is extremely simple to extend in gecGEC - just change the inner beta distribution to a k-ary dDirichlet distribution. aA full explanation is beyond the scope of this post, so iI'll explain this and details of other extensions in future blog posts.

iIn addition, gecGEC can enhance traditional or bBayesian modeling. gecGEC acts as a wrapper layer rather than the model itself - one could attach models or model elements to gecGEC propositions, and reason about belief on those propositions. oOften, model estimates come with hidden assumptions. tThese can be investigated using gecGEC, and may aid with choosing the best model.

sSpeaking of choosing the best model, causal inference often involves proposing several potential models and seeing which fit the evidence. tThis is perfect for gecGEC; in fact, unlike with numeric modeling, gecGEC can be integrated closely with causal diagrams. gecGEC's relational propositions become causal propositions, and a causal engine can identify backdoor paths, confounding, bias, and more.