your objection is similar to proposing zeno's paradox and then saying no no we can't solve that using
calculus, we weren't talking about
math. you want a system formal enough to say things like "law of noncontradiction" and force a binary, but not so formal that it becomes answerable. which is to say, an argument merely of convenience.
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Gödel's theorems show limits of formal systems doing math, not of reasoning itself. Euthyphro's logic is philosophical reasoning, not a formal system vulnerable to Gödel's limits.
lol no, this is one of the silliest things youve ever said. philosophical reasoning fundamentally relies on logic to structure arguments and assure coherence, and to derive conclusions systematically. Gödel's theorems show the limits to axiomatic reasoning and that can apply to any philosophical system with sufficient expressiveness. dismissing this ignores how logic underpins the statements you want to use.
but that's fine. let's actually do the exercise.
let's create a moral system E for Euthyphro, call it MSE.
we'll use constants that represent specific actions (a1, a2 etc) and natural numbers (0, 1, 2)
it has predicates, such as G(a) meaning action a is good, D(a) meaning action a is commanded by the gods, and I(a) meaning action a is good according to an independent standard.
we'll use arithmetic symbols to represent things like addition (+) equality (=) or a successor function S(n) = n + 1 along with logical connectives, quantifiers, and variables to note different actions or numbers (like x, y, etc).
this system has a syntax, like moral statements, arithmetic statements, and the combination of them.
put together three groups of axioms: moral axioms M for the dilemma itself, arithmetic axioms A, and consistency axiom C. we'll call them M1, M2... A1, A2... etc
M1 states that for all x(G(x) if and only if (D(x) or I(x))) --- meaning an action is good if and only if it is either commanded by the gods or good by an independent standard. this is the dilemma's binary for two possible sources of goodness.
M2 states there does not exist x ((D(x) and I(x)) --- meaning no action is both commanded by the gods and good by an independent standard. this is mutual exclusivity - can't be both derived from command and the independent standard - no overlap.
M3 states there exists x G(x) -- meaning at least one good action does in fact exist. this is makes the system nontrivial such as morality exists.
M4 states that for all x (D(x) -> G(x)) --- meaning if an action is commanded by the gods, it is good. this is a tentative, because it may conflict with M5 which says...
M5 for all x (I(x) -> G(h)) -- meaning if an action is good by an independent standard of good, then it is good. M4 and M5 together may lead to inconsistency - the whole crux of the matter. it is possible that M4 and M5 must be taken as competing hypothesis to avoid contradictions, i haven't considered this long enough to know if i have enough constraints here.
Alright, then we have arithmetic axioms.
A1: for all x ( S(x) =! 0) --- the successor for any number is not zero
A2: for all x and for all y (S(x) = S(y) -> x = y) --- if the successors of two numbers are equal, then the numbers themselves are equal
A3: for all x (x+0=x) --- adding zero to a number yields the same number
A4: for all x and for all y (x+S(y) = S(x+y)) -- addition is defined recursively via the successor meaning x+(y+1) is the same as (x+y)+1
A5: for all x (x*0=0) --- any number times zero is zero
A6: for all x and for all y (x*S(y) = (x*y) + x) --- multiplication is defined recursively
and finally consistency axiom C1: the system is consistent
the system MSE follows inference rules:
modus ponens (or the rule of detachment) where if P and P -> Q, then Q
universal generalization: if P(x) is provable for an arbitrary x, then for all x P(x)
and standard logical rules (not, and, or, therefore, equivalent, exists)
now, within that framework we can take the first incompleteness theorem. P: this statement is not provable in MSE. If P is true, it is not provable (as it claims). If P is false, it is provable, which is a contradiction since a false statement being provable is an inconsistency. if we assume consistency, P is true but not provable.
within MSE itself, if we construct a statement Q such that:
Q is for every x (G(x) <-> exists y) where y is a fundamental moral principle
here goodness is defined by a fundamental moral principle, neither divine nor independent but intrinsic. this could be unprovable in MSE, where "good" is an axiom outside of the system's completeness.
the second incompleteness theorem would be that MSE cannot prove its own consistency. so if we say "MSE does not prove a contradiction" that is unprovable within MSE.
applying this to Euthyphro's dilemma, we would posit "good" as a fundamental axiom within MSE, unprovable within the system but nevertheless true - which is NEITHER of the options presented (divine command, or independent standard).
Axiom G: there exists x G(x) AND for all x (G(x) -> not (D(x) <-> I(x))) --- meaning there exists a good action, and goodness is not reducible to either divine command or an independent standard alone.
This statement is true within MSE, but is unprovable within the system.
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So again, if we're going to play around with logic, unless you can find a problem with the logic above, "good" can
logically be an unprovable axiom which resolves the dilemma by making good intrinsic, not derived from either divine command OR the independent standard. and similarly, if morality is not a formal system then the dilemma is similarly false, because "good" can emerge non-axiomatically from context or intuition.
the dilemma's basic assumption is that of a logical, axiomatic structure which invite's Gödel's analysis, and the unprovable nature of good within that structure defeats the binary choice presented. this removes the tension between arbitrariness and subordination. either way the dilemma is false.
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The argument that morality can arise from context or intuition is merely choosing one horn of the dilemma or rejecting good as objectively defined. Neither of which is problematic for me.
it literally is not. i invite you to prove it.
