I got an email from Fr. John Rickert today, and with his permission I’ll share part of it here.
A sin of commission occurs when we do something we should not do. A system is consistent (or maybe I should say “sound”) if the results of proofs really are true. Gödel’s 2nd Incompleteness Theorem says that it is undecidable whether Peano Arithmetic commits any “sins of commission.”
A sin of omission occurs when we fail to do something that we should do. A system is complete if every true statement actually has a proof (in finitely many steps). Gödel’s 1st Incompleteness Theorem says that Peano Arithmetic does commit some “sins of omission”: There are truths that cannot be proved.
Finally, a conscience is perplexed if it does not know whether to do or refrain from a proposed action; the conscience is de facto in a state of invincible ignorance. Undecidability is invincible ignorance.
Of course a formal system isn’t under any moral obligations, and certainly not under obligation to do what it cannot do. These are just analogies. But they are interesting analogies. Sins of commission and omission, things done and things left undone, are more verbally parallel than completeness and soundness.
Here’s another post based on an email exchange with Fr. Rickert exactly one year ago: Unexpected square wave.
Lamport’s ideas about specification. There are safety specifications (that say what say what state-changes mustn’t ever happen), and “liveness” specifications. I hesitate to say what liveness specs are, lest I incur his wrath, … oh well, why not. There are (at least) two types. One is *weak* fairness that says that if some state-change is eventually always enabled, it must eventually occur. The other is *strong* fairness, it is sufficient that the state-change is infinitely often enabled.
So if you had permanently the chance to help the Samaritan, but never did, you sin against weak fairness. Whereas, if you only get the chance infinitely often, but never did, you sin against strong (stricter) fairness.
These can be thought about topologically. A safety specification is a closed subset of the “Baire” space of sequences of (discrete) states. I have to admit I don’t recall exactly what the fairness conditions mean topologically. I vaguely recall they both have something to do with density.
A start to chase down Lamport’s ideas about fairness/liveness might (maybe) be found here: https://lamport.azurewebsites.net/pubs/lamport-fairness.pdf
Thanks for posting, John. Sins of commission and sins of omission are certainly something I relate to concretely; the analogy helps me get a better grasp of what the Incompleteness Theorems really mean.
Deontic logic employs the machinery of modal logic to formalize ethical reasoning. The symbol that indicates “necessary” is re-interpreted as “obligatory” and “possible” becomes “permissible”.
Am perplexed, albeit at a humbler level:
Wouldn’t be a “sin of commission“ be asserting (proving) a false result? In other words, if the Principia Mathematica could prove a statement and its opposite — that would be a sin of commission. But Gödel only goes so far as to say that there are statements which have neither proofs nor disproofs in PM, not that any sins are committed. (The Second Incompleteness theorem says that there’s no way for PM to prove of the consistency of PM, not that PM is, in fact, inconsistent.)
A “sin of omission” might be a failure to assert (prove) something that’s true — Certainly Gödel says PM (and any related or stronger system ) commits such sins. “This Truth has no proof.” However for some systems, consistency might indeed be proved in a larger superset of that system (what this means morally and ethically is murky.)
Perplexity indeed!!
A question. It seems to me that if one were to prove contradictory statements, then one would prove that the system (e.g., Peano Arithmetic) is inconsistent, which Goedel’s Second Theorem says cannot be done. May we therefore proceed with the assumption that PA is consistent, similarly to the way we deal with vacuously true statements?
@Fr John- (my understanding) One may show the consistency of PA as subset, using an axiomatic superset containing PA — but then in turn that larger system cannot prove its own consistency.
Thus one climbs a ladder of invincible ignorance, where the lower rungs are solid but all present and future rungs are unknowably secure.